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2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE " " -1 -1 "Helvetica" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 38 "MATRICES AND MATRIX OPE RATIONS: Unit 1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 387 23 "Dr. Wlodzislaw Kostecki" }}{PARA 259 " " 0 "" {TEXT -1 53 "The Papua New Guinea University of Technology (PNG UT)" }}{PARA 259 "" 0 "" {TEXT -1 54 "Department of Electrical and Com munication Engineering" }}{PARA 259 "" 0 "" {TEXT -1 20 "Lae, Morobe P rovince" }}{PARA 259 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 386 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 388 19 "All \+ rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 593 67 "-------------------------------------------------------- -----------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 821 3 "(1)" }{TEXT 823 1 " " }{TEXT 822 12 "Introduction" } {TEXT -1 0 "" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 824 10 "OBJECTIVES" }{TEXT 825 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 826 1 "\225" } {TEXT -1 98 " To provide a collection of the necessary definitions of , and concepts associated with, matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 827 1 "\225" }{TEXT -1 62 " To present notational conventions a dopted for all the Units." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 828 1 "\225 " }{TEXT -1 15 " To introduce " }{TEXT 833 5 "Maple" }{TEXT 832 1 " \222" }{TEXT -1 18 "s library package " }{TEXT 834 6 "linalg" }{TEXT -1 53 " containing functions required for matrix operations." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 846 1 "\225" }{TEXT -1 34 " To introduce the concepts of a " }{TEXT 847 20 "one-dimensional list" }{TEXT -1 6 " or " }{TEXT 848 5 "array" }{TEXT -1 7 " and " }{TEXT 849 20 "t wo-dimensional list" }{TEXT -1 6 " or " }{TEXT 850 13 "list of lists " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 865 1 "\225" }{TEXT -1 15 " To introduce " }{TEXT 867 5 "Maple" }{TEXT 866 1 "\222" } {TEXT -1 25 "s type-checking function " }{TEXT 868 4 "type" }{TEXT -1 10 " and the " }{TEXT 871 7 "Boolean" }{TEXT -1 10 " values " } {XPPEDIT 18 0 "true;" "6#%%trueG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "false;" "6#%&falseG" }{TEXT -1 17 " returned by it." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 829 1 "\225" }{TEXT -1 70 " To show alternative methods of defining and inputting matrices with " }{TEXT 835 5 "Maple " }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 830 1 "\225" }{TEXT -1 97 " To illustrate methods of creating matrices whose elements dep end on their location in a matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 831 1 "\225" }{TEXT -1 71 " To show alternative methods of substituti ng data for matrix elements." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 836 1 " \225" }{TEXT -1 155 " To show alternative methods of evaluation of ma trices entered with symbolic elements whose numerical values become kn own in a later phase of computation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 837 1 "\225" }{TEXT -1 28 " To introduce the rule of " }{TEXT 839 20 "last-name evaluation" }{TEXT -1 10 " used by " }{TEXT 840 5 "Mapl e" }{TEXT -1 56 " as applied to matrices and illustrate its implicatio ns." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 838 1 "\225" }{TEXT -1 47 " To s pecify the functions from outside of the " }{TEXT 842 6 "linalg" } {TEXT -1 33 " package, which are contained in " }{TEXT 844 5 "Maple" } {TEXT 843 1 "\222" }{TEXT -1 3 "s " }{TEXT 845 4 "main" }{TEXT -1 57 " library and are able to perform operations on matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 841 1 "\225" }{TEXT -1 79 " To present alternat ive methods of unassigning a matrix structure from a name." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 634 3 "1 .1" }{TEXT 636 1 " " }{TEXT 635 44 "Basic definitions and notational c onventions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "A matrix is an ordered or systematic " }{TEXT 856 1 "(" }{TEXT -1 11 "rectangular" }{TEXT 858 1 ")" }{TEXT -1 2 " \+ " }{TEXT 857 5 "array" }{TEXT -1 6 " of " }{TEXT 291 7 "scalars" } {TEXT -1 11 ", called " }{TEXT 381 8 "elements" }{TEXT -1 26 ", whi ch are arranged in " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 12 " row s and " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 98 " columns and encl osed in brackets. Such a matrix is said to be of order (or size, or di mension) " }{TEXT 548 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 290 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 549 1 ")" }{TEXT -1 2 " " }{TEXT 550 1 "(" }{TEXT 855 7 "read: " }{XPPEDIT 18 0 "m" "6#%\"m G" }{TEXT -1 6 " by " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 551 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The elements of a matrix are double-subsc ripted to denote their location in the matrix, the " }{TEXT 705 3 "ro w" }{TEXT -1 9 " index " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 17 " preceding the " }{TEXT 706 6 "column" }{TEXT -1 9 " index " } {XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 26 ". The subscript indices " }{TEXT 19 4 "must" }{TEXT -1 6 " be " }{TEXT 707 15 "natural numbers " }{TEXT -1 90 ", which may or may not be separated by the comma \226 refer to Section 1.2 for more details." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "The " }{TEXT 382 7 "scalars" }{TEXT -1 10 " may be " }{TEXT 383 12 "real numbers" } {TEXT -1 3 ", " }{TEXT 384 15 "complex numbers" }{TEXT -1 7 ", or \+ " }{TEXT 385 9 "functions" }{TEXT -1 20 " of some parameter." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "A matrix is " }{TEXT 371 11 "real-valued" }{TEXT -1 2 " " } {TEXT 851 1 "(" }{TEXT -1 4 "or " }{TEXT 372 4 "real" }{TEXT -1 1 " \+ " }{TEXT 852 1 ")" }{TEXT -1 6 " if " }{TEXT 373 3 "all" }{TEXT -1 20 " its elements are " }{TEXT 374 12 "real numbers" }{TEXT -1 6 " \+ or " }{TEXT 375 21 "real-valued functions" }{TEXT -1 16 ". A matrix \+ is " }{TEXT 376 14 "complex-valued" }{TEXT -1 2 " " }{TEXT 853 1 "( " }{TEXT -1 4 "or " }{TEXT 377 7 "complex" }{TEXT -1 1 " " }{TEXT 854 1 ")" }{TEXT -1 15 " if at least " }{TEXT 378 3 "one" }{TEXT -1 16 " element is a " }{TEXT 379 14 "complex number" }{TEXT -1 8 " or a " }{TEXT 380 23 "complex-valued function" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If " }{TEXT 596 3 "all" }{TEXT -1 28 " elements of a matrix ar e " }{TEXT 597 7 "numbers" }{TEXT -1 12 ", whether " }{TEXT 598 4 " real" }{TEXT -1 6 " or " }{TEXT 599 7 "complex" }{TEXT -1 24 ", the n it is called a " }{TEXT 600 8 "constant" }{TEXT -1 14 " matrix. If " }{TEXT 788 3 "all" }{TEXT -1 28 " elements of a matrix are " } {TEXT 789 8 "integers" }{TEXT -1 25 ", then it is called an " } {TEXT 790 7 "integer" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Two matrices ar e " }{TEXT 389 5 "equal" }{TEXT -1 73 " if they have the same order \+ and their corresponding elements are equal." }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "A matrix is " }{TEXT 292 6 "square" }{TEXT -1 97 " if the number of rows is equal t o the number of columns. Such a matrix is said to be of order " } {TEXT 552 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 569 3 " \327 " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 553 1 ")" }{TEXT -1 6 " or " } {TEXT 554 1 "(" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 570 3 " \327 " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 555 1 ")" }{TEXT -1 66 ". A square matrix whose associated determinant is non-vanishing " }{TEXT 859 1 "(" }{TEXT -1 8 "is not " }{XPPMATH 20 "6#%%zeroG" }{TEXT 860 1 ")" } {TEXT -1 15 " is called a " }{TEXT 293 12 "non-singular" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A matrix containing only " }{TEXT 294 3 "one" }{TEXT -1 22 " column is called a " }{TEXT 295 13 "column matrix" } {TEXT 296 2 ". " }{TEXT -1 15 " Its order is " }{TEXT 556 1 "(" } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 297 3 " \327 " }{XPPEDIT 18 0 "1" " 6#\"\"\"" }{TEXT 557 1 ")" }{TEXT -1 29 ". A matrix containing only \+ " }{TEXT 298 3 "one" }{TEXT -1 19 " row is called a " }{TEXT 299 10 "row matrix" }{TEXT -1 17 ". Its order is " }{TEXT 558 1 "(" } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 300 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 559 1 ")" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 301 4 "N.B." }{TEXT -1 27 " In textbooks, the names " }{TEXT 302 13 "column vector" }{TEXT -1 7 " and " }{TEXT 303 10 "row vector" }{TEXT -1 33 " are used int erchangeably with " }{TEXT 304 13 "column matrix" }{TEXT -1 7 " and \+ " }{TEXT 305 10 "row matrix" }{TEXT -1 20 ", respectively. In " } {TEXT 309 5 "Maple" }{TEXT -1 7 ", the " }{TEXT 306 6 "vector" } {TEXT -1 8 " is a " }{TEXT 307 3 "row" }{TEXT -1 16 " object and is " }{TEXT 308 3 "not" }{TEXT -1 51 " a matrix, and the concept of \"co lumn vector\" does " }{TEXT 310 3 "not" }{TEXT -1 51 " exist at all \+ \226 refer to Unit (3) for more details." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Operations on mat rices are performed using functions and commands contained in a " } {TEXT 637 5 "Maple" }{TEXT -1 24 " library package called " }{TEXT 311 6 "linalg" }{TEXT -1 2 " (" }{TEXT 312 3 "lin" }{TEXT -1 4 "ear " }{TEXT 313 3 "alg" }{TEXT -1 40 "ebra package), which is loaded with t he " }{TEXT 314 4 "with" }{TEXT -1 37 " command. The package includes \+ over " }{XPPEDIT 18 0 "100" "6#\"$+\"" }{TEXT -1 27 " functions and \+ procedures." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 638 12 "with(linalg)" }{TEXT -1 111 " command is used in every Unit in such a way that the functions n ecessary for a given Unit are specified, e.g. " }{TEXT 639 44 "with(li nalg, det, inverse, multiply, rowdim)" }{TEXT -1 27 ". This results in loading " }{TEXT 640 4 "only" }{TEXT -1 24 " a few functions, not \+ " }{TEXT 641 3 "all" }{TEXT -1 77 " the contents of the package. It s hould be noted that although the function " }{TEXT 683 6 "matrix" } {TEXT -1 16 " belongs to the " }{TEXT 682 6 "linalg" }{TEXT -1 75 " pa ckage, it need not be specified since this function is also included i n " }{TEXT 685 5 "Maple" }{TEXT 684 1 "\222" }{TEXT -1 3 "s " }{TEXT 686 4 "main" }{TEXT -1 10 " library." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 343 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 344 7 "NOTE 1:" }{TEXT -1 120 " In textbooks, a matrix is frequently \+ designated by a single boldface capital letter often enclosed in brack ets, e.g. [" }{TEXT 345 1 "A" }{TEXT -1 81 "]. Wherever possible, the \+ same notation is used in the text of all the Units. In " }{TEXT 346 5 "Maple" }{TEXT -1 183 ", a matrix is identified by a non-bracketed nam e, e.g. A. Consequently, in commands found in the Units, most often ca pitals are used to denote matrices, e.g. A, M, U, V, Z, Y, CM, RM." }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Where " }{TEXT 393 9 "functions" }{TEXT -1 6 " of " }{TEXT 595 8 "constant" }{TEXT -1 63 " matrices are involved, such matrices \+ are denoted in text as " }{TEXT 560 4 "cos(" }{TEXT -1 1 "[" }{TEXT 645 1 "A" }{TEXT -1 1 "]" }{TEXT 561 2 ")," }{TEXT -1 2 " " }{TEXT 643 3 "ln(" }{TEXT -1 1 "[" }{TEXT 642 1 "A" }{TEXT -1 1 "]" }{TEXT 644 2 ")," }{TEXT -1 14 " etc. Where " }{TEXT 603 9 "functions" } {TEXT -1 25 " of matrices comprising " }{TEXT 601 10 " functions" } {TEXT -1 22 " of a real variable " }{TEXT 602 1 "t" }{TEXT -1 46 " \+ are involved, such matrices are denoted as " }{TEXT 651 4 "sin(" } {TEXT -1 1 "[" }{TEXT 647 1 "A" }{TEXT 649 1 "(" }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT 648 1 ")" }{TEXT -1 1 "]" }{TEXT 650 2 ")," }{TEXT -1 2 " " }{TEXT 646 4 "exp(" }{TEXT -1 1 "[" }{TEXT 652 1 "A" }{TEXT 654 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 653 1 ")" }{TEXT -1 1 "] " }{TEXT 655 2 ")," }{TEXT -1 132 " etc. If mathematical expressions \+ of functions of matrices are involved in text, then the following matr ix designations are used: " }{TEXT 656 4 "tan(" }{TEXT 657 1 "A" } {TEXT 658 2 ")," }{TEXT -1 2 " " }{TEXT 659 7 "sinh(A(" }{TEXT 660 1 "t" }{TEXT 661 3 "))," }{TEXT -1 6 " etc." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Where a " }{TEXT 394 8 "function" }{TEXT -1 72 " is applied to all elements of a matri x, such matrices are denoted as " }{TEXT 562 5 "A_exp" }{TEXT -1 3 ", " }{TEXT 563 4 "A_ln" }{TEXT -1 3 ", " }{TEXT 564 5 "A_sin" }{TEXT -1 3 ", " }{TEXT 565 6 "A_sqrt" }{TEXT -1 7 ", etc." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 342 7 "N OTE 2:" }{TEXT -1 109 " Various notational methods are found in textb ooks for the names of certain matrices, using the characters " } {TEXT 337 1 "'" }{TEXT -1 2 " " }{TEXT 861 1 "(" }{TEXT -1 4 "or " } {XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 18 " as a superscript " }{TEXT 862 1 ")" }{TEXT -1 7 " and " }{TEXT 338 1 "*" }{TEXT -1 2 " " }{TEXT 863 1 "(" }{TEXT -1 4 "or " }{TEXT 339 1 "t" }{TEXT -1 21 " as a subscript or " }{TEXT 591 1 "T" }{TEXT -1 18 " as a super script" }{TEXT 864 1 ")" }{TEXT -1 17 " to denote the " }{TEXT 340 7 "inverse" }{TEXT -1 11 " and the " }{TEXT 341 9 "transpose" } {TEXT -1 134 " of a matrix, respectively. For all the Units, the foll owing names are adopted for unification and convenience in displaying \+ results:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 484 1 "\225" }{TEXT -1 2 " \+ " }{TEXT 486 4 "Abs(" }{TEXT 487 1 "A" }{TEXT 488 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 485 14 "absolute value" }{TEXT -1 31 " of t he elements of a matrix [" }{TEXT 489 1 "A" }{TEXT -1 2 "]," }}} {EXCHG {PARA 0 "" 0 "" {TEXT 347 1 "\225" }{TEXT -1 2 " " }{TEXT 348 4 "Adj(" }{TEXT 490 1 "A" }{TEXT 491 1 ")" }{TEXT -1 14 " \226 for t he " }{TEXT 363 7 "adjoint" }{TEXT -1 22 " of a square matrix [" } {TEXT 492 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 349 1 "\225" }{TEXT -1 2 " " }{TEXT 350 4 "Arg(" }{TEXT 494 1 "A" }{TEXT 495 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 364 18 "principal ar gument" }{TEXT -1 31 " of the elements of a matrix [" }{TEXT 493 1 "A " }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 465 1 "\225" } {TEXT -1 2 " " }{TEXT 466 8 "char_eq(" }{TEXT 497 1 "A" }{TEXT 498 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 467 23 "characteristic equ ation" }{TEXT -1 22 " of a square matrix [" }{TEXT 496 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 453 1 "\225" }{TEXT -1 2 " \+ " }{TEXT 454 10 "char_poly(" }{TEXT 500 1 "A" }{TEXT 501 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 455 25 "characteristic polynomial" } {TEXT -1 22 " of a square matrix [" }{TEXT 499 1 "A" }{TEXT -1 2 "], " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 456 1 "\225" }{TEXT -1 2 " " } {TEXT 457 11 "char_roots(" }{TEXT 504 1 "A" }{TEXT 505 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 458 20 "characteristic roots" }{TEXT -1 6 " or " }{TEXT 459 11 "eigenvalues" }{TEXT -1 22 " of a square \+ matrix [" }{TEXT 502 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 460 1 "\225" }{TEXT -1 2 " " }{TEXT 461 13 "char_vectors(" } {TEXT 506 1 "A" }{TEXT 507 1 ")" }{TEXT -1 14 " \226 for the " } {TEXT 462 22 "characteristic vectors" }{TEXT -1 6 " or " }{TEXT 463 12 "eigenvectors" }{TEXT -1 22 " of a square matrix [" }{TEXT 503 1 " A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 351 1 "\225" } {TEXT -1 2 " " }{TEXT 352 9 "Cofactor(" }{XPPEDIT 18 0 "a[ij]" "6#&% \"aG6#%#ijG" }{TEXT 509 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 365 8 "cofactor" }{TEXT -1 17 " of an element " }{XPPEDIT 18 0 "a[ij ]" "6#&%\"aG6#%#ijG" }{TEXT -1 22 " of a square matrix [" }{TEXT 508 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 510 1 "\225" } {TEXT -1 2 " " }{TEXT 511 9 "Cofactor(" }{TEXT 514 1 "A" }{TEXT 515 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 512 15 "cofactor matrix " }{TEXT -1 35 " associated with a square matrix [" }{TEXT 513 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 353 1 "\225" }{TEXT -1 2 " " }{TEXT 354 5 "Conj(" }{TEXT 522 1 "A" }{TEXT 523 1 ")" } {TEXT -1 14 " \226 for the " }{TEXT 366 9 "conjugate" }{TEXT -1 23 " of a complex matrix [" }{TEXT 516 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 1 "\225" }{TEXT -1 2 " " }{TEXT 356 4 "Det( " }{TEXT 524 1 "A" }{TEXT 525 1 ")" }{TEXT -1 14 " \226 for the " } {TEXT 367 11 "determinant" }{TEXT -1 22 " of a square matrix [" } {TEXT 517 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 357 1 "\225" }{TEXT -1 2 " " }{TEXT 358 4 "Inv(" }{TEXT 526 1 "A" }{TEXT 527 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 368 7 "inverse" } {TEXT -1 35 " of a square non-singular matrix [" }{TEXT 518 1 "A" } {TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 359 1 "\225" }{TEXT -1 2 " " }{TEXT 360 6 "Minor(" }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ij G" }{TEXT 532 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 369 5 "min or" }{TEXT -1 17 " of an element " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6 #%#ijG" }{TEXT -1 22 " of a square matrix [" }{TEXT 519 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 541 1 "\225" }{TEXT -1 2 " \+ " }{TEXT 542 18 "roots_and_vectors(" }{TEXT 546 1 "A" }{TEXT 547 1 ") " }{TEXT -1 43 " \226 for the sequence of lists containing " } {TEXT 543 20 "characteristic roots" }{TEXT -1 7 " and " }{TEXT 544 7 "vectors" }{TEXT -1 22 " of a square matrix [" }{TEXT 545 1 "A" } {TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 450 1 "\225" }{TEXT -1 2 " " }{TEXT 451 6 "Trace(" }{TEXT 530 1 "A" }{TEXT 531 1 ")" } {TEXT -1 14 " \226 for the " }{TEXT 452 5 "trace" }{TEXT -1 22 " o f a square matrix [" }{TEXT 520 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 361 1 "\225" }{TEXT -1 2 " " }{TEXT 362 7 "Tran sp(" }{TEXT 528 1 "A" }{TEXT 529 1 ")" }{TEXT -1 14 " \226 for the \+ " }{TEXT 370 9 "transpose" }{TEXT -1 15 " of a matrix [" }{TEXT 521 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Moreover, the following auxiliary \+ names are adopted:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 776 1 "\225" } {TEXT -1 2 " " }{TEXT 777 8 "Antisym(" }{TEXT 780 1 "A" }{TEXT 781 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 778 18 "antisymmetric part " }{TEXT -1 22 " of a square matrix [" }{TEXT 779 1 "A" }{TEXT -1 15 "] defined as \275" }{TEXT 799 1 "(" }{TEXT -1 1 "[" }{TEXT 797 1 "A " }{TEXT -1 2 "] " }{TEXT 802 1 "\226" }{TEXT -1 1 " " }{TEXT 801 6 "T ransp" }{TEXT -1 1 "[" }{TEXT 798 1 "A" }{TEXT -1 1 "]" }{TEXT 800 1 " )" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 782 1 "\225" } {TEXT -1 2 " " }{TEXT 783 4 "Sym(" }{TEXT 786 1 "A" }{TEXT 787 1 ")" }{TEXT -1 14 " \226 for the " }{TEXT 784 14 "symmetric part" } {TEXT -1 22 " of a square matrix [" }{TEXT 785 1 "A" }{TEXT -1 15 "] \+ defined as \275" }{TEXT 793 1 "(" }{TEXT -1 1 "[" }{TEXT 791 1 "A" } {TEXT -1 2 "] " }{TEXT 794 1 "+" }{TEXT -1 1 " " }{TEXT 796 6 "Transp " }{TEXT -1 1 "[" }{TEXT 792 1 "A" }{TEXT -1 1 "]" }{TEXT 795 1 ")" } {TEXT -1 1 "," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 420 1 "\225" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Col[j]" "6#&%$ColG6#%\"jG" }{TEXT 539 1 "(" } {TEXT 535 1 "A" }{TEXT 536 1 ")" }{TEXT -1 14 " \226 for the " } {XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 24 "th column of a matrix [" } {TEXT 533 1 "A" }{TEXT -1 2 "]," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 421 1 "\225" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Row[i]" "6#&%$RowG6#%\"iG" } {TEXT 540 1 "(" }{TEXT 537 1 "A" }{TEXT 538 1 ")" }{TEXT -1 14 " \226 for the " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 21 "th row of a m atrix [" }{TEXT 534 1 "A" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 5 "(1.2)" }{TEXT 396 1 " " }{TEXT 395 59 "Entering matrices and substituting data for m atrix elements" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "restart : with(linalg, adjoint, d et, entermatrix, orthog, randmatrix, rowdim, transpose) :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "In " }{TEXT 268 5 "Maple" }{TEXT -1 156 ", there are several alternative methods of entering matrices and substituting data for matrix element s. An overview of these methods is presented hereunder." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 403 1 "A" }{TEXT -1 2 ". " }{TEXT 287 31 "Defining and inputting matrices" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 566 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 567 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 568 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 404 1 "A" }{TEXT -1 31 "] containing symbolic elements." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 266 6 "matrix" }{TEXT -1 102 " function, type th e matrix elements in brackets containing bracketed elements of each co nsecutive row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "A := matr ix([ [a[11], a[12], a[13]], [a[21], a[22], a[23]] ]) : A = matrix (A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6 #\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 408 4 "N.B." } {TEXT -1 61 " Equivalent to the above short command is defining first a " }{TEXT 422 20 "two-dimensional list" }{TEXT -1 6 " or " } {TEXT 423 13 "list of lists" }{TEXT -1 7 ", i.e." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 73 "LL := [ [a[11], a[12], a[13]], [a[21], a[2 2], a[23]] ] : 'LL' = LL ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#LL G7$7%&%\"aG6#\"#6&F(6#\"#7&F(6#\"#87%&F(6#\"#@&F(6#\"#A&F(6#\"#B" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and then, converting it into a matrix using the " }{TEXT 426 7 "convert" }{TEXT -1 5 " and " }{TEXT 427 6 "matrix" }{TEXT -1 16 " fun ctions, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A := conver t(LL, matrix) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"# A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 315 6 "matrix" }{TEXT -1 52 " function, specify the number of matrix rows \+ (e.g., " }{TEXT 319 1 "2" }{TEXT -1 43 ") followed by the number of co lumns (e.g., " }{TEXT 320 1 "3" }{TEXT -1 60 ") and then, type the mat rix elements in brackets row-by-row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "A := matrix(2, 3, [a[11], a[12], a[13], a[21], a[22], a[23]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG -%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6 #\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 409 4 "N.B." }{TEXT -1 61 " Equivalent to the above short co mmand is defining first a " }{TEXT 424 20 "one-dimensional list" } {TEXT -1 6 " or " }{TEXT 425 5 "array" }{TEXT -1 7 ", i.e." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "L := [a[11], a[12], a[13], a [21], a[22], a[23]] : 'L' = L ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"LG7(&%\"aG6#\"#6&F'6#\"#7&F'6#\"#8&F'6#\"#@&F'6#\"#A&F'6#\"#B" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and then, converting it into a matrix using the " }{TEXT 428 6 "matrix" }{TEXT -1 15 " function, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A := matrix(2, 3, L) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6 #\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 8 "Method 3" }{TEXT -1 12 ". Using the " }{TEXT 317 5 "array" }{TEXT -1 101 " function, type the array elements in brackets containing bracketed elements of each cons ecutive row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A := array( [ [a[11], a[12], a[13]], [a[21], a[22], a[23]] ]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6#\" #6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 410 4 "N.B." } {TEXT -1 61 " Equivalent to the above short command is defining first a " }{TEXT 433 20 "two-dimensional list" }{TEXT -1 6 " or " } {TEXT 434 13 "list of lists" }{TEXT -1 7 ", i.e." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 73 "LL := [ [a[11], a[12], a[13]], [a[21], a[2 2], a[23]] ] : 'LL' = LL ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#LL G7$7%&%\"aG6#\"#6&F(6#\"#7&F(6#\"#87%&F(6#\"#@&F(6#\"#A&F(6#\"#B" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and then, converting it into a matrix using the " }{TEXT 435 7 "convert" }{TEXT -1 5 " and " }{TEXT 436 6 "matrix" }{TEXT -1 16 " fun ctions, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A := conver t(LL, matrix) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"# A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 8 "Method 4" }{TEXT -1 12 ". Using the " }{TEXT 271 5 "array" }{TEXT -1 50 " function, specify the range of array rows (e. g., " }{TEXT 318 4 "1..2" }{TEXT -1 42 ") followed by the range of col umns (e.g., " }{TEXT 322 4 "1..3" }{TEXT -1 102 ") and then, type the \+ array elements in brackets containing bracketed elements of each conse cutive row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "A := array(1 ..2, 1..3, [ [a[11], a[12], a[13]], [a[21], a[22], a[23]] ]) : A \+ = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7 %&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 411 4 "N.B." }{TEXT -1 61 " Equivalent to the above short command is \+ defining first a " }{TEXT 429 20 "two-dimensional list" }{TEXT -1 6 " or " }{TEXT 430 13 "list of lists" }{TEXT -1 7 ", i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "LL := [ [a[11], a[12], a[13]], [a [21], a[22], a[23]] ] : 'LL' = LL ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#LLG7$7%&%\"aG6#\"#6&F(6#\"#7&F(6#\"#87%&F(6#\"#@&F(6#\"#A&F(6 #\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and then, converting it into a matrix using the " } {TEXT 431 7 "convert" }{TEXT -1 5 " and " }{TEXT 432 6 "matrix" } {TEXT -1 16 " functions, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A := convert(LL, matrix) : A = matrix(A) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87 %&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 8 "Method 5" }{TEXT -1 12 ". Using the " }{TEXT 323 7 "convert" }{TEXT -1 151 " function, type the eleme nts in brackets containing bracketed elements of each consecutive row \+ and then, convert this two-dimensional structure into a " }{TEXT 324 6 "matrix" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "A := convert([ [a[11], a[12], a[13]], [a[21], a[22], a[23]] ], \+ matrix) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG- %'matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6# \"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 412 4 "N.B." }{TEXT -1 61 " Equivalent to the above short com mand is defining first a " }{TEXT 437 20 "two-dimensional list" } {TEXT -1 6 " or " }{TEXT 438 13 "list of lists" }{TEXT -1 7 ", i.e. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "LL := [ [a[11], a[12], a[13]], [a[21], a[22], a[23]] ] : 'LL' = LL ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%#LLG7$7%&%\"aG6#\"#6&F(6#\"#7&F(6#\"#87%&F(6#\"#@&F (6#\"#A&F(6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and then, converting it into a matrix usi ng the " }{TEXT 439 7 "convert" }{TEXT -1 5 " and " }{TEXT 440 6 "matr ix" }{TEXT -1 16 " functions, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A := convert(LL, matrix) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%&%\"aG6#\"#6&F+6 #\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 3 "B. " }{TEXT 405 37 "Substituting data for matrix elements" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For example, c onsider " }{TEXT 571 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 572 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 573 1 ")" }{TEXT -1 12 " \+ matrices [" }{TEXT 398 1 "A" }{TEXT -1 4 "], [" }{TEXT 272 1 "B" } {TEXT -1 8 "], and [" }{TEXT 273 1 "V" }{TEXT -1 8 "], and " }{TEXT 574 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 575 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 576 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 474 1 "E" }{TEXT -1 17 "] to illustrate " }{TEXT 325 15 "non -interactive" }{TEXT -1 16 " methods, and " }{TEXT 577 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 578 3 " \327 " }{XPPEDIT 18 0 "2" " 6#\"\"#" }{TEXT 579 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 274 1 "F" } {TEXT -1 21 "] to illustrate the " }{TEXT 326 11 "interactive" } {TEXT -1 9 " method." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 327 8 "Method 1" }{TEXT -1 36 ". Assign v alues to matrix elements " }{TEXT 259 8 "prior to" }{TEXT -1 19 " de fining a matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "a[11]:=2 : a[12]:=1 : a[13]:=0 : a[21]:=1 : a[22]:=-1 : a[23]:=3 : a[31]:=0 : \+ a[32]:=6 : a[33]:=2 :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "and then, define a matrix of desir ed size using the same symbolic names for its elements:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A := matrix([[a[11], a[12], a[13]] , [a[21], a[22], a[23]], [a[31], a[32], a[33]]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\"#\"\"\"\" \"!7%F+!\"\"\"\"$7%F,\"\"'F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 401 4 "N.B." }{TEXT -1 147 " This met hod is useful where a set of data is to be presented in the matrix for m. To this end, type first the data in a manner indicating that a " } {TEXT 468 33 "two-dimensional ordered structure" }{TEXT -1 19 " is in volved, i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "data_list: =[ [a, b, c], [d, e, f] ] : 'data_list'=data_list ; type(data_list, li st) ; type(data_list, listlist) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %*data_listG7$7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "This is a " }{TEXT 441 20 "two-dimensional list" }{TEXT -1 7 " , or " }{TEXT 402 13 "list of lists" }{TEXT -1 24 ", as verified by both " }{TEXT 872 7 "Boolean" }{TEXT -1 10 " values " }{XPPMATH 20 "6#%%trueG" }{TEXT -1 41 " returned by the type-checking function \+ " }{TEXT 406 4 "type" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Then, convert the two-d imensional list (list of lists) into a matrix using the " }{TEXT 442 7 "convert" }{TEXT -1 5 " and " }{TEXT 443 6 "matrix" }{TEXT -1 16 " f unctions, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := conv ert(data_list, matrix) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%%\"aG%\"bG%\"cG7%%\"dG%\"eG%\"fG " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 397 8 "Method 2" }{TEXT -1 46 ". Declare the name and dimensions of a matrix," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "B := matri x(3, 3) :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 17 "or, equivalently," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "B := matrix(3, 3, [ ]) : B := array(1..3, 1..3) : \+ B := array(1..3, 1..3, [ ]) :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "assign values to its elements ," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "B[1,1]:=-2 : B[1,2]:= 3 : B[1,3]:=1 : B[2,1]:=0 : B[2,2]:=3 : B[2,3]:=0 : B[3,1]:=4 : B[3,2] :=5 : B[3,3]:=2 :" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "and input/display the matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "B := matrix(B) : B = matrix(B) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%!\"#\"\"$\" \"\"7%\"\"!F+F.7%\"\"%\"\"&\"\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 399 74 "Notice the difference in \+ the way of assigning values to matrix elements in" }{TEXT -1 13 " Met hods 1 " }{TEXT 400 3 "and" }{TEXT -1 5 " 2 " }{TEXT 604 57 "if a c apital letter is used as the name of a matrix, viz." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 605 1 "\225" } {TEXT -1 5 " in " }{TEXT 774 8 "Method 1" }{TEXT -1 3 ": " }{TEXT 606 20 "lower-case letters " }{TEXT 19 4 "must" }{TEXT 609 76 " be u sed for the symbolic indexed matrix elements and the index digits may \+ " }{TEXT 607 3 "not" }{TEXT 608 27 " be separated by the comma." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 610 1 "\225" }{TEXT -1 5 " in " }{TEXT 775 8 "Method 2" }{TEXT -1 3 " : " }{TEXT 611 20 "upper-case letters " }{TEXT 19 4 "must" }{TEXT 613 64 " be used for the symbolic indexed matrix elements and the com ma" }{TEXT -1 2 " " }{TEXT 19 4 "must" }{TEXT -1 2 " " }{TEXT 612 25 "separate the index digits" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 621 44 "Also notice a t this point that matrix names " }{TEXT 620 6 "cannot" }{TEXT 622 140 " be subscripted in the way, which is used in indexing variables, i.e. an attempt to assign a matrix structure to single-indexed names like \+ " }{TEXT 614 4 "B[1]" }{TEXT 623 2 " " }{TEXT 615 1 "(" }{XPPEDIT 18 0 "B[1]" "6#&%\"BG6#\"\"\"" }{TEXT 616 1 ")" }{TEXT 624 6 " or " }{TEXT 617 4 "B[a]" }{TEXT 625 2 " " }{TEXT 618 1 "(" }{XPPEDIT 18 0 "B[a]" "6#&%\"BG6#%\"aG" }{TEXT 619 1 ")" }{TEXT 626 81 " will result in an error message. Indexing matrix names follows different rules." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 809 58 "Two cases of indexed matrix names should be distinguishe d." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 803 1 "\225" }{TEXT -1 2 " " }{TEXT 804 6 "Case 1" }{TEXT -1 1 " " }{TEXT 810 59 " when an indexed name is to be assigned a matrix st ructure." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 806 18 "A matrix name may " }{TEXT 805 4 "only" }{TEXT 807 42 " be added a subscript, which consists of " }{TEXT 19 3 "two" }{TEXT 808 259 " natural numbers separated by the comma, neither of t he numbers being greater than the number of rows or columns, respectiv ely, of a matrix structure assigned to the subscripted name. For examp le, the following indexed name and matrix assignment are correct:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "B[2,3] := matrix(2, 3, [b[11 ], b[12], b[13], b[21], b[22], b[23]]) : B[2,3] = matrix(B[2,3]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6$\"\"#\"\"$-%'matrixG6#7$7%& %\"bG6#\"#6&F/6#\"#7&F/6#\"#87%&F/6#\"#@&F/6#\"#A&F/6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 811 1 " \225" }{TEXT -1 2 " " }{TEXT 812 6 "Case 2" }{TEXT -1 1 " " }{TEXT 813 25 " when an indexed name is " }{TEXT 818 3 "not" }{TEXT 819 35 " \+ to be assigned a matrix structure." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 815 18 "A matrix name may " } {TEXT 814 4 "only" }{TEXT 816 42 " be added a subscript, which consist s of " }{TEXT 19 3 "two" }{TEXT 820 35 " strings separated by the co ma and" }{TEXT -1 1 " " }{TEXT 817 126 "formed by enclosing any sequen ce of characters in a pair of backquotes. For instance, the following \+ indexed names may be used:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "B[`(11`,`23)`] = matrix(2, 3, [b[11], b[12], b[13], b[21], b[22], \+ b[23]]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6$%$(11G%$23)G-%'m atrixG6#7$7%&%\"bG6#\"#6&F/6#\"#7&F/6#\"#87%&F/6#\"#@&F/6#\"#A&F/6#\"# B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 86 "B[`(2 rows`,`3 columns)`] = matrix(2, 3, [b[11], b[ 12], b[13], b[21], b[22], b[23]]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"BG6$%((2~rowsG%+3~columns)G-%'matrixG6#7$7%&%\"bG6#\"#6&F/6#\"#7 &F/6#\"#87%&F/6#\"#@&F/6#\"#A&F/6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 328 8 "Method 3" }{TEXT -1 55 ". Define a matrix using symbolic names for its elements" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "V := array(1..3, 1..3, [[v[1 1], v[12], v[13]], [v[21], v[22], v[23]], [v[31], v[32], v[33]]]) :" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "then use the " }{TEXT 329 4 "subs" }{TEXT -1 13 " functio n to " }{TEXT 330 4 "subs" }{TEXT -1 33 "titute the symbolic names in \+ the " }{TEXT 331 6 "matrix" }{TEXT -1 62 " function with the given val ues, and input/display the matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "V := subs(v[11]=8, v[12]=2, v[13]=1, v[21]=1, v[22]= 3, v[23]=0, v[31]=4, v[32]=2, v[33]=1, matrix(V)) : V =matrix(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"VG-%'matrixG6#7%7%\"\")\"\"#\"\" \"7%F,\"\"$\"\"!7%\"\"%F+F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 469 8 "Method 4" }{TEXT -1 88 ". If a m atrix containing functions of symbolic elements has been defined and i nput, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "E := matrix( 2, 3, [(3*a-2*b)^2, 1/4*exp(b/a), -2*tan(2*a/(a^2+b^2)), 2*cos(a/b), 4 /7*sinh(2*a^2+b), 3/5*a^(-sin(b))]) : E = matrix(E) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7%*$),&%\"aG\"\"$*&\"\"#\"\" \"%\"bGF1!\"\"F0F1,$-%$expG6#*&F2F1F-F3#F1\"\"%,$-%$tanG6#,$*&F-F1,&*$ )F-F0F1F1*$)F2F0F1F1F3F0!\"#7%,$-%$cosG6#*&F-F1F2F3F0,$-%%sinhG6#,&FBF 0F2F1#F:\"\"(,$)F-,$-%$sinG6#F2F3#F.\"\"&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "and numerical da ta for the symbolic elements are determined and input in a later phase of computations, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a := 1.75 : b := -0.25 : 'a' = a ; 'b' = b ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG$\"$v\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"bG$!#D!\"#" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "then floating-point evaluation of matrix [" } {TEXT 762 1 "E" }{TEXT -1 66 "] may be performed using any of the foll owing alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "(a) Using the function " }{TEXT 763 3 "map" }{TEXT -1 52 " together with the arrow-type functional ope rator, " }{TEXT 766 2 "( " }{TEXT 765 3 "x->" }{TEXT 767 2 " )" } {TEXT -1 20 ", and the function " }{TEXT 764 4 "eval" }{TEXT -1 1 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "E = map(x->eval(x), E) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7%$\"'D1L!\"% $\"+]Z>n@!#5$!+A0\">8%!\"*7%$\"+4X!y]\"F2$\"+\"H0s,\"!\"($\"+S:'4*oF/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7%$\"'D1L!\"%$ \"+]Z>n@!#5$!+A0\">8%!\"*7%$\"+4X!y]\"F2$\"+\"H0s,\"!\"($\"+S:'4*oF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7%$\"'D1L!\"%$\" +]Z>n@!#5$!+A0\">8%!\"*7%$\"+4X!y]\"F2$\"+\"H0s,\"!\"($\"+S:'4*oF/" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "(b) Using the functions " }{TEXT 771 5 "evalf" }{TEXT -1 5 " an d " }{TEXT 768 4 "subs" }{TEXT -1 36 " together with any of the functi ons " }{TEXT 769 4 "eval" }{TEXT -1 2 ", " }{TEXT 770 6 "matrix" } {TEXT -1 4 ",or " }{TEXT 772 2 "op" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "E = evalf(subs('a'=a, 'b'=b, eval(E))) ; \+ E = evalf(subs('a'=a, 'b'=b, matrix(E))) ; E = evalf(subs('a'=a, 'b '=b, op(E))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$ 7%$\"'D1L!\"%$\"+]Z>n@!#5$!+A0\">8%!\"*7%$\"+4X!y]\"F2$\"+\"H0s,\"!\"( $\"+S:'4*oF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7% $\"'D1L!\"%$\"+]Z>n@!#5$!+A0\">8%!\"*7%$\"+4X!y]\"F2$\"+\"H0s,\"!\"($ \"+S:'4*oF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"EG-%'matrixG6#7$7%$ \"'D1L!\"%$\"+]Z>n@!#5$!+A0\">8%!\"*7%$\"+4X!y]\"F2$\"+\"H0s,\"!\"($\" +S:'4*oF/" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 44 "Notice the single unevaluation quotes that " } {TEXT 19 4 "must" }{TEXT -1 21 " surround each data " }{TEXT 773 4 "n ame" }{TEXT -1 15 " in method (b)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 333 8 "Method 5" }{TEXT -1 14 ". This is an " }{TEXT 332 11 "interactive" }{TEXT -1 25 " method, w hich uses the " }{TEXT 269 11 "entermatrix" }{TEXT -1 132 " function p rompting the user to enter the number of element values equal to the n umber of symbolic elements in a pre-defined matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 444 6 "Step 1 " }{TEXT -1 56 ". Define a matrix using symbolic names for its element s:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG-%'matrixG6#7$7$&%\"fG6#\" #6&F+6#\"#77$&F+6#\"#@&F+6#\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "F := matrix(2, 2, [f[11], f[12], f[21], f[22]]) : F = matrix (F) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG-%'matrixG6#7$7$&%\"fG6 #\"#6&F+6#\"#77$&F+6#\"#@&F+6#\"#A" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 445 6 "Step 2" }{TEXT -1 59 ". Enter values (numerical or symbolic) of the elements of [" }{TEXT 334 1 "F" }{TEXT -1 27 "] given, for instance, as " }{XPPEDIT 18 0 "f [11]=-2" "6#/&%\"fG6#\"#6,$\"\"#!\"\"" }{TEXT 470 1 "," }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "f[12]=3" "6#/&%\"fG6#\"#7\"\"$" }{TEXT 471 1 "," } {TEXT -1 2 " " }{XPPEDIT 18 0 "f[21]=4" "6#/&%\"fG6#\"#@\"\"%" } {TEXT 472 1 "," }{TEXT -1 2 " " }{XPPEDIT 18 0 "f[22]=6" "6#/&%\"fG6# \"#A\"\"'" }{TEXT 473 1 "." }{TEXT -1 22 " The values entered " } {TEXT 19 4 "must" }{TEXT -1 35 " be terminated with the semi-colon" } {TEXT 407 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := ente rmatrix(F) :" }}}{EXCHG {PARA 0 "enter element 1,1 > " 0 "" {MPLTEXT 1 0 3 "-2;" }}}{EXCHG {PARA 0 "enter element 1,2 > " 0 "" {MPLTEXT 1 0 2 "3;" }}}{EXCHG {PARA 0 "enter element 2,1 > " 0 "" {MPLTEXT 1 0 2 "4;" }}}{EXCHG {PARA 0 "enter element 2,2 > " 0 "" {MPLTEXT 1 0 2 "6; " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 446 6 "Step 3" }{TEXT -1 57 ". Input/display the matrix containi ng the values entered:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F = matrix(F) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG-%'matrixG6#7$ 7$!\"#\"\"$7$\"\"%\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 275 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 390 4 "N.B." }{TEXT -1 76 " Matrices whose elements are to be a function of the element l ocation, i.e." }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "a[ij] = f(i, j)" "6#/&%\"aG6#%#ijG-%\"fG6$%\"iG%\"jG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "may be created us ing either of the following alternative methods." }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For exampl e, create a " }{TEXT 580 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 581 3 " \327 " }{XPPEDIT 18 0 "5" "6#\"\"&" }{TEXT 582 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 391 1 "A" }{TEXT -1 42 "] whose elements are given by the function" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "a[i j] = 2*i^3 - 3*j^2" "6#/&%\"aG6#%#ijG,&*&\"\"#\"\"\"*$%\"iG\"\"$F+F+*& F.F+*$%\"jGF*F+!\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 413 8 "Method 1" }{TEXT -1 19 ". Using th e double " }{TEXT 464 3 "for" }{TEXT -1 16 "-loop construct:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "A := array(1..4, 1..5) : f or i to 4 do for j to 5 do A[i,j] := 2*i^3 - 3*j^2 : od : od :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := matrix(A) : A = ma trix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7&7'!\" \"!#5!#D!#Y!#t7'\"#8\"\"%!#6!#K!#f7'\"#^\"#U\"#F\"\"'!#@7'\"$D\"\"$;\" \"$,\"\"#!)\"#`" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 414 8 "Method 2" }{TEXT -1 25 ". Using the speci al call " }{TEXT 415 17 "matrix(m, n, fnc)" }{TEXT -1 5 " and " } {TEXT 745 5 "Maple" }{TEXT 744 1 "\222" }{TEXT -1 40 "s arrow-type def inition of the function " }{TEXT 628 3 "fnc" }{TEXT -1 26 " acting on \+ the locations " }{TEXT 629 1 "(" }{TEXT 630 1 "i" }{TEXT 632 1 "," } {TEXT 633 2 " j" }{TEXT 631 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "fnc := (i, j) -> 2*i^3 - 3*j^2 : A := matrix(4, 5, \+ fnc) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'m atrixG6#7&7'!\"\"!#5!#D!#Y!#t7'\"#8\"\"%!#6!#K!#f7'\"#^\"#U\"#F\"\"'!# @7'\"$D\"\"$;\"\"$,\"\"#!)\"#`" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 258 "" 0 "" {TEXT 417 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 416 4 "N.B." } {TEXT -1 6 " If " }{TEXT 418 3 "all" }{TEXT -1 125 " the symbolic o r numerical elements of a pre-defined and input matrix are to be subst ituted with (overwritten by) the same " }{TEXT 419 7 "integer" } {TEXT -1 18 ", then using the " }{TEXT 475 3 "map" }{TEXT -1 5 " and \+ " }{TEXT 476 2 "op" }{TEXT -1 91 " functions is the simplest way. For \+ instance, substitute all elements of the above matrix [" }{TEXT 482 1 "A" }{TEXT -1 20 "] with the integer " }{XPPEDIT 18 0 "12" "6#\"#7" } {TEXT 743 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A := map( op(12), A) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\" AG-%'matrixG6#7&7'\"#7F*F*F*F*F)F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 480 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 481 4 "N .B." }{TEXT -1 6 " If " }{TEXT 479 3 "all" }{TEXT -1 119 " the symb olic or numerical elements of a matrix are to be substituted with (ove rwritten by) an arbitrary non-integer " }{TEXT 447 4 "real" }{TEXT -1 6 " or " }{TEXT 448 14 "complex number" }{TEXT -1 9 ", some " } {TEXT 449 9 "symbolic " }{TEXT -1 14 " element, or " }{TEXT 662 8 "fu nction" }{TEXT -1 44 " of a real variable or complex number, the " } {TEXT 477 3 "map" }{TEXT -1 61 " function together with the arrow-type functional operator, " }{TEXT 583 2 "( " }{TEXT 478 3 "x->" }{TEXT 584 2 " )" }{TEXT -1 78 ", should be used. For instance, substitute a ll elements of the above matrix [" }{TEXT 483 1 "A" }{TEXT -1 34 "] wi th the result of the product " }{XPPEDIT 18 0 "2*I*cos(1.27)" "6#*(\" \"#\"\"\"%\"IGF%-%$cosG6#$\"$F\"!\"#F%" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A := map(x -> 2*I*cos(1.27), A) : \+ A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6# 7&7'^#$\"+euhDf!#5F*F*F*F*F)F)F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 392 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 4 "N.B." } {TEXT -1 37 " For creating example matrices, the " }{TEXT 276 10 "ran dmatrix" }{TEXT -1 41 " function may be used, which returns an " } {TEXT 585 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 278 3 " \327 " } {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 586 1 ")" }{TEXT -1 48 " matrix wi th random integers from the interval " }{XPPEDIT 18 0 "[-99...99]" "6# 7#;,$\"#**!\"\"F&" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For example, create a " } {TEXT 587 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 588 3 " \327 " } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 589 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 279 1 "R" }{TEXT -1 22 "] with random entries." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "R := randmatrix(3, 4) : R = matri x(R) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"RG-%'matrixG6#7%7&!#&)!# b!#P!#N7&\"#(*\"#]\"#z\"#c7&\"#\\\"#j\"#d!#f" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "An optional arg ument may be used together with the " }{TEXT 280 10 "randmatrix" } {TEXT -1 81 " function, which will create a special type of random mat rix. The option may be " }{TEXT 282 8 "diagonal" }{TEXT -1 3 ", " } {TEXT 283 9 "symmetric" }{TEXT -1 3 ", " }{TEXT 284 13 "antisymmetric " }{TEXT -1 3 ", " }{TEXT 285 6 "sparse" }{TEXT -1 3 ", " }{TEXT 288 10 "unimodular" }{TEXT -1 7 ", or " }{TEXT 286 8 "identity" } {TEXT -1 68 ". Refer to Units (7) and (9) for the definitions of thes e matrices." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 675 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If " }{TEXT 674 9 "variation " }{TEXT -1 67 " of constants given in the form of symbolic elements \+ of a matrix [" }{TEXT 673 1 "P" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "P := matrix(2, 3, [G, H, J, K, N, T]) : P = \+ matrix(P) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"PG-%'matrixG6#7$7%% \"GG%\"HG%\"JG7%%\"KG%\"NG%\"TG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "is to be done, i.e. each ele ment is to be made a function of a single variable" }{TEXT 676 1 "," } {TEXT -1 2 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 677 1 "," }{TEXT -1 44 " then the following command should be used:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "P := P(x) : 'P(x)' = matrix(P) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"PG6#%\"xG-%'matrixG6#7$7%-%\"GGF& -%\"HGF&-%\"JGF&7%-%\"KGF&-%\"NGF&-%\"TGF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 678 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "A matrix [" }{TEXT 679 1 "M" }{TEXT -1 53 "] whose elements are to be a function of a variable " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 680 1 "," }{TEXT -1 89 " which function is to depend on the ele ment location in the matrix may be created using " }{TEXT 704 5 "Maple " }{TEXT 703 1 "\222" }{TEXT -1 40 "s arrow-type definition of the fun ction " }{TEXT 746 3 "fnc" }{TEXT -1 22 " and the special call " } {TEXT 681 17 "matrix(m, n, fnc)" }{TEXT -1 6 ", e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "fnc := (i, j) -> (3*i - 2*j)*t^(2*i + 3*j ) : M(t) := matrix(2, 3, fnc) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> -%\"MG6#%\"tG-%'matrixG6#7$7%*$)F'\"\"&\"\"\",$*$)F'\"\")F0!\"\",$*$)F '\"#6F0!\"$7%,$*$)F'\"\"(F0\"\"%,$*$)F'\"#5F0\"\"#\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 281 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 4 "N.B." }{TEXT -1 96 " Unlike in computations where n ames retain the assigned objects (values, functions, etc.), in " } {TEXT 263 6 "matrix" }{TEXT -1 47 " computations the name assigned a \+ matrix does " }{TEXT 261 3 "not" }{TEXT -1 98 " \"carry over\" the str ucture assigned. This means that if the name V has been assigned the m atrix [" }{TEXT 264 1 "V" }{TEXT -1 70 "] (as is the case earlier in t his Unit), then typing in the name does " }{TEXT 267 3 "not" }{TEXT -1 13 " return the " }{TEXT 335 6 "matrix" }{TEXT -1 11 " but the \+ " }{TEXT 336 8 "variable" }{TEXT -1 36 " name to which it is assigned , i.e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "V ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%\"VG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "This special rule used by " } {TEXT 663 5 "Maple" }{TEXT -1 49 " in the evaluation of data structure s including " }{TEXT 687 6 "arrays" }{TEXT -1 7 " and " }{TEXT 688 8 "matrices" }{TEXT -1 13 " is called " }{TEXT 664 20 "last-name eva luation" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Consequently, any operation appli ed to a matrix name returns a result of a given operation performed on the " }{TEXT 665 8 "variable" }{TEXT -1 31 " used as the matrix nam e, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "V + 1 ; V^2 ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"VG\"\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"VG\"\"#\"\"\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "There are, howeve r, some functions that do not belong to the " }{TEXT 666 6 "linalg" } {TEXT -1 139 " package but will recognise the pre-defined matrix struc ture assigned to the name and perform the respective operations on the matrix, viz." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 724 1 "\225" }{TEXT -1 11 " Function " }{TEXT 723 4 "copy" }{TEXT -1 33 " that duplicates data structures:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "copy(V) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F*\"\"$\"\"!7%\"\"% F)F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 726 1 "\225" }{TEXT -1 11 " Function " }{TEXT 725 7 "entries " }{TEXT -1 81 " that displays an unordered sequence of the bracketed \+ entries of data structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "entries(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+7#\"\")7#\"\"#7#\" \"\"F'7#\"\"%7#\"\"$F'7#\"\"!F%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 728 1 "\225" }{TEXT -1 11 " Functi on " }{TEXT 727 5 "evalm" }{TEXT -1 6 " that " }{TEXT 729 4 "eval" } {TEXT -1 6 "uates " }{TEXT 730 1 "m" }{TEXT -1 77 "atrices and express ions involving matrices both symbolically and numerically:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F*\"\"$\"\"!7%\"\"%F) F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 732 1 "\225" }{TEXT -1 11 " Function " }{TEXT 731 7 "indices" } {TEXT -1 96 " that displays an unordered sequence of the bracketed ind ices of the entries of data structures:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "indices(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+7$\" \"\"F$7$F$\"\"#7$F$\"\"$7$F&F$7$F(F$7$F&F&7$F(F(7$F&F(7$F(F&" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Although the above four functions are not included in the " } {TEXT 667 6 "linalg" }{TEXT -1 80 " package, each of them is specifica lly designed to work on data structures only." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Some other com mands and operators from " }{TEXT 757 5 "Maple" }{TEXT 756 1 "\222" } {TEXT -1 3 "s " }{TEXT 758 4 "main" }{TEXT -1 142 " library that rec ognise the matrix structure \"behind\" its name and perform the respec tive operation on the pre-defined matrix are as follows." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 753 1 " \225" }{TEXT -1 11 " Function " }{TEXT 754 7 "convert" }{TEXT -1 42 " together with a suitable parameter (e.g. " }{TEXT 755 8 "listlist" } {TEXT -1 46 ") that converts an expression to another form:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "convert(V, listlist) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%7%\"\")\"\"#\"\"\"7%F'\"\"$\"\"!7%\" \"%F&F'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 708 1 "\225" }{TEXT -1 11 " Function " }{TEXT 709 5 "print " }{TEXT -1 68 " that displays the values of the expressions appearing as arguments:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(V) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\")\"\"#\"\"\"7 %F*\"\"$\"\"!7%\"\"%F)F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 710 1 "\225" }{TEXT -1 17 " Type-checkin g " }{TEXT 712 7 "Boolean" }{TEXT -1 11 " function " }{TEXT 711 4 "t ype" }{TEXT -1 81 " that checks if the first parameter is of type spec ified by the second parameter:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "type(V, matrix) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 719 1 "\225" }{TEXT -1 11 " Function " }{TEXT 720 3 "map" } {TEXT -1 57 " used together with the arrow-type functional operator, \+ " }{TEXT 748 2 "( " }{TEXT 747 3 "x->" }{TEXT 749 2 " )" }{TEXT -1 202 ", that applies a function or procedure to all the operands (comp onents) of expressions, including matrices. In the example below, the \+ function applied to V is multiplication of each element of matrix [" } {TEXT 722 1 "V" }{TEXT -1 17 "] by the scalar " }{XPPEDIT 18 0 "mu" " 6#%#muG" }{TEXT 721 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(x->x*mu, V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7% ,$%#muG\"\"),$F)\"\"#F)7%F),$F)\"\"$\"\"!7%,$F)\"\"%F+F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 741 1 " \225" }{TEXT -1 96 " A pair of empty parentheses placed immediately a fter the matrix name that displays the matrix:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "V() ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7%7%\"\")\"\"#\"\"\"7%F*\"\"$\"\"!7%\"\"%F)F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 733 1 "\225" } {TEXT -1 66 " Subscript notation, or bracketed indices separated by t he comma " }{TEXT 734 6 "[i, j]" }{TEXT -1 38 " that extracts the elem ent of matrix [" }{TEXT 735 1 "V" }{TEXT -1 29 "] at the specified loc ation " }{TEXT 736 1 "(" }{TEXT 737 1 "i" }{TEXT 738 2 ", " }{TEXT 739 1 "j" }{TEXT 740 2 ")," }{TEXT -1 7 " e.g.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "V[3,1] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 759 1 "\225" }{TEXT -1 11 " Function " }{TEXT 760 9 "conjugat e" }{TEXT -1 120 " that computes the complex conjugate of complex-numb ered expressions, including matrices. If a complex-numbered matrix [" }{TEXT 761 1 "Z" }{TEXT -1 13 "] is given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Z := matrix(2, 3, [-5*I, 2+6*I, I, 4+8*I, 3, 6-1 0*I]) : Z = matrix(Z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"ZG-%' matrixG6#7$7%^#!\"&^$\"\"#\"\"'^#\"\"\"7%^$\"\"%\"\")\"\"$^$F.!#5" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "then" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "conjugate(Z) \+ = evalm(conjugate(Z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*conjuga teG6#%\"ZG-%'matrixG6#7$7%^#\"\"&^$\"\"#!\"'^#!\"\"7%^$\"\"%!\")\"\"$^ $\"\"'\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "There are two commands in " }{TEXT 751 5 "Maple" }{TEXT 750 1 "\222" }{TEXT -1 3 "s " }{TEXT 752 4 "main" }{TEXT -1 89 " library that force full evaluation of a matrix name to the eleme nts of the matrix, viz." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 713 1 "\225" }{TEXT -1 11 " Function " } {TEXT 714 4 "eval" }{TEXT -1 12 " that fully " }{TEXT 718 4 "eval" } {TEXT -1 38 "uates expressions, including matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F*\"\"$\"\"!7%\"\"%F)F*" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 715 1 "\225" }{TEXT -1 11 " Function " }{TEXT 716 2 "op" }{TEXT -1 15 " that extracts " }{TEXT 717 2 "op" }{TEXT -1 131 "erands (componen ts) of expressions, including matrices. For the latter, the function r eturns all the components in the matrix form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "op(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7%7%\"\")\"\"#\"\"\"7%F*\"\"$\"\"!7%\"\"%F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Needles s to say that all the functions and commands contained in the " } {TEXT 668 6 "linalg" }{TEXT -1 113 " package recognise the matrix stru cture \"behind\" its name and perform the relevant operations on the m atrix, e.g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "adjoint(V) \+ ; det(V) ; orthog(V) ; rowdim(V) ; transpose(V) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"$\"\"!!\"$7%!\"\"\"\"%\"\"\" 7%!#5!\")\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\")\"\"\"\"\"%7% \"\"#\"\"$F,7%F)\"\"!F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "To \"inform\" a function from outs ide of the " }{TEXT 669 6 "linalg" }{TEXT -1 91 " package that the nam e V has been assigned a matrix structure, the aforementioned commands \+ " }{TEXT 670 4 "eval" }{TEXT -1 5 " and " }{TEXT 671 2 "op" }{TEXT -1 13 " or function " }{TEXT 672 6 "matrix" }{TEXT -1 18 " may be used, e .g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eval(V) + 1 ; (op( V))^2 ; 3/4*matrix(V) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'matr ixG6#7%7%\"\")\"\"#\"\"\"7%F+\"\"$\"\"!7%\"\"%F*F+F+F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*$)-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F,\"\"$\"\" !7%\"\"%F+F,F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%'matrixG6#7%7% \"\")\"\"#\"\"\"7%F+\"\"$\"\"!7%\"\"%F*F+#F-F0" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "[ How to exe cute the above operations, refer to the pertinent Units. ]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "I n summary, using any of the following commands will re-assign the matr ix structure to the name V:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "V := copy(V) : V := evalm(V) : V := eval(V) : V := op(V) : \+ V := matrix(V) : V := V() ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" VG-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F,\"\"$\"\"!7%\"\"%F+F," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 742 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "If a matrix name is to be unassigned the structure of a pre-defined matrix, any of the following alternative m ethods may be used." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 689 8 "Method 1" }{TEXT -1 22 ". Using th e procedure " }{TEXT 690 8 "unassign" }{TEXT -1 6 ", viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "type(V, matrix) ; unassign(V) : \+ V ; type(V , matrix) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"VG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 691 8 "Method 2" }{TEXT -1 23 ". Using single qu otes " }{TEXT 694 2 "( " }{TEXT 692 1 "'" }{TEXT -1 1 " " }{TEXT 693 1 "'" }{TEXT 695 2 " )" }{TEXT -1 33 " enclosing the matrix name, viz ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "type(E, matrix) ; E \+ := 'E' : E ; type(E , matrix) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"EG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 699 8 "Method 3" }{TEXT -1 12 ". Using th e " }{TEXT 700 5 "evaln" }{TEXT -1 15 " function that " }{TEXT 701 4 " eval" }{TEXT -1 28 "uates the expression to its " }{TEXT 702 1 "n" } {TEXT -1 9 "ame, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ty pe(R, matrix) ; R := evaln(R) : R ; type(R , matrix) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% \"RG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The first \+ " }{TEXT 869 7 "Boolean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%%tr ueG" }{TEXT -1 100 " verifies that a name carries the pre-defined mat rix structure before unassigning, and the second " }{TEXT 870 7 "Bool ean" }{TEXT -1 9 " value " }{XPPMATH 20 "6#%&falseG" }{TEXT -1 55 " \+ verifies that the name has been unassigned the matrix" }{TEXT -1 11 " structure." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 289 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 590 4 "N.B." }{TEXT -1 26 " A matri x name of type [" }{TEXT 696 1 "M" }{TEXT 698 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 697 1 ")" }{TEXT -1 147 "] retains the structure a ssigned. To display such a name together with a pre-defined matrix str ucture, enclose it in a pair of single quotes, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "'M(t)' = M(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"MG6#%\"tG-%'matrixG6#7$7%*$)F'\"\"&\"\"\",$*$)F'\" \")F0!\"\",$*$)F'\"#6F0!\"$7%,$*$)F'\"\"(F0\"\"%,$*$)F'\"#5F0\"\"#\"\" !" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 " " {TEXT 627 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (2) for \"" } {TEXT 594 33 "Extracting components of matrices" }{TEXT -1 2 "\"." }}} {EXCHG {PARA 258 "" 0 "" {TEXT 592 67 "------------------------------- ------------------------------------" }}}}{MARK "251 0 0" 0 } {VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }