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University of Technology (PNGUT)" }} {PARA 275 "" 0 "" {TEXT -1 54 "Department of Electrical and Communicat ion Engineering" }}{PARA 276 "" 0 "" {TEXT -1 20 "Lae, Morobe Province " }}{PARA 277 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 " " {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT 426 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 279 "" 0 "" {TEXT 428 19 "All rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 280 "" 0 "" {TEXT 429 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 264 3 "(3)" }{TEXT 347 1 " " }{TEXT 346 36 "Addition and subtr action of matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 450 10 "OBJECTIVES" }{TEXT 451 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 452 1 " \225" }{TEXT -1 77 " To define the operations of matrix addition and \+ subtraction and state the " }{TEXT 468 28 "addition conformability ru le" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 453 1 "\225" } {TEXT -1 73 " To provide alternative methods of matrix addition and s ubtraction with " }{TEXT 454 5 "Maple" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 455 1 "\225" }{TEXT -1 64 " To specify the laws obeyed by matrix addition and subtraction." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 456 1 "\225" }{TEXT -1 31 " To introduce the concept of " } {TEXT 457 5 "equal" }{TEXT -1 11 " matrices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 458 1 "\225" }{TEXT -1 24 " To show how to add a " }{TEXT 459 6 "scalar" }{TEXT -1 15 " and a matrix." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 460 1 "\225" }{TEXT -1 24 " To show how to add a " }{TEXT 461 6 "scalar" }{TEXT -1 30 " to each element of a matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 462 1 "\225" }{TEXT -1 29 " To show how \+ to obtain with " }{TEXT 463 5 "Maple" }{TEXT -1 91 " the correct resul t of subtraction of a matrix from itself and introduce the concept of \+ a " }{TEXT 471 4 "zero" }{TEXT -1 6 " or " }{TEXT 472 4 "null" } {TEXT -1 9 " matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart : with(linalg, dia g, matadd) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 318 1 "A" }{TEXT -1 2 ". " }{TEXT 274 15 "Matrix add ition" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The matrices to be added " }{TEXT 19 4 "must" }{TEXT -1 12 " obey the " }{TEXT 257 28 "addition conformability rule" } {TEXT -1 16 ", which states:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "\"Two matrices can only be ad ded together only if they are both of the " }{TEXT 258 4 "same" } {TEXT -1 9 " order.\"" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Such matrices are called " } {TEXT 323 11 "conformable" }{TEXT -1 6 " or " }{TEXT 324 10 "compati ble" }{TEXT -1 35 " for addition. If the matrices do " }{TEXT 325 3 " not" }{TEXT -1 33 " satisfy this rule, their sum is " }{TEXT 321 3 "no t" }{TEXT -1 15 " defined (does " }{TEXT 322 3 "not" }{TEXT -1 8 " exi st)." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 33 "If two matrices to be added are [" }{TEXT 259 1 "A" } {TEXT -1 12 "] of order " }{TEXT 356 1 "(" }{XPPEDIT 357 0 "m" "6#%\" mG" }{TEXT 358 3 " \327 " }{XPPEDIT 359 0 "n" "6#%\"nG" }{TEXT 360 1 " )" }{TEXT -1 17 " with elements " }{XPPEDIT 18 0 "a[ij]" "6#&%\"aG6# %#ijG" }{TEXT -1 7 " and [" }{TEXT 260 1 "B" }{TEXT -1 12 "] of order " }{TEXT 361 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 362 3 " \327 \+ " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 363 1 ")" }{TEXT -1 17 " with e lements " }{XPPEDIT 18 0 "b[ij]" "6#&%\"bG6#%#ijG" }{TEXT 464 1 "," } {TEXT -1 17 " then the sum [" }{TEXT 261 1 "A" }{TEXT -1 2 "] " } {TEXT 263 1 "+" }{TEXT -1 2 " [" }{TEXT 262 1 "B" }{TEXT -1 18 "] is \+ the matrix [" }{TEXT 465 1 "C" }{TEXT -1 12 "] of order " }{TEXT 364 1 "(" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 365 3 " \327 " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 366 1 ")" }{TEXT -1 17 " with elements " } {XPPEDIT 18 0 "c[ij]" "6#&%\"cG6#%#ijG" }{TEXT -1 47 " that are found from the following definition." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The element of the " } {XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 12 "th row and " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 14 "th column of [" }{TEXT 466 1 "C" }{TEXT -1 46 "] is obtained by summing the elements of the " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 16 "th row and the " }{XPPEDIT 18 0 "j" "6#% \"jG" }{TEXT -1 27 "th column of both matrices." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Thus, the ge neral element of the sum matrix [" }{TEXT 467 1 "C" }{TEXT -1 13 "] is given by" }}}{EXCHG {PARA 283 "" 0 "" {XPPEDIT 18 0 "c[ij]=a[ij]+b[ij ]" "6#/&%\"cG6#%#ijG,&&%\"aG6#F'\"\"\"&%\"bG6#F'F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For examp le, consider a " }{TEXT 447 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 448 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 449 1 ")" } {TEXT -1 10 " matrix [" }{TEXT 446 1 "A" }{TEXT -1 10 "] given as" }} }{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$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\" #87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "and a " }{TEXT 367 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 368 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 369 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 265 1 "B" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 " B := matrix(2, 3, [b[11], b[12], b[13], b[21], b[22], b[23]]) : B = \+ matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7%& %\"bG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "T he sum [" }{TEXT 307 1 "A" }{TEXT -1 2 "] " }{TEXT 309 1 "+" }{TEXT -1 2 " [" }{TEXT 308 1 "B" }{TEXT -1 9 "] is a " }{TEXT 370 1 "(" } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 371 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 372 1 ")" }{TEXT -1 82 " matrix, which may be obtaine d using either of the following alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 348 8 "Method \+ 1" }{TEXT -1 12 ". Using the " }{TEXT 349 6 "matadd" }{TEXT -1 10 " fu nction:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`A+B` := matadd( A, B) : A + B = matrix(`A+B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ ,&%\"AG\"\"\"%\"BGF&-%'matrixG6#7$7%,&&%\"aG6#\"#6F&&%\"bGF0F&,&&F/6# \"#7F&&F3F6F&,&&F/6#\"#8F&&F3F;F&7%,&&F/6#\"#@F&&F3FAF&,&&F/6#\"#AF&&F 3FFF&,&&F/6#\"#BF&&F3FKF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 350 8 "Method 2" }{TEXT -1 12 ". Using th e " }{TEXT 351 5 "evalm" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "`A+B` := evalm(A + B) : A + B = matrix(`A+ B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"%\"BGF&-%'matri xG6#7$7%,&&%\"aG6#\"#6F&&%\"bGF0F&,&&F/6#\"#7F&&F3F6F&,&&F/6#\"#8F&&F3 F;F&7%,&&F/6#\"#@F&&F3FAF&,&&F/6#\"#AF&&F3FFF&,&&F/6#\"#BF&&F3FKF&" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This matrix addition may be displayed in \"like-in-a-book\" for m, namely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "matrix(A) + ma trix(B) = matrix(`A+B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'mat rixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\"#B \"\"\"-F&6#7$7%&%\"bGF,&FDF/&FDF27%&FDF6&FDF9&FDF-F&6#7$7%,&F*F>FCF >,&F.F>FEF>,&F1F>FFF>7%,&F5F>FHF>,&F8F>FIF>,&F;F>FJF>" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 310 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 4 "N.B." }{TEXT -1 22 " Matrix addition is " }{TEXT 328 11 "commutative" }{TEXT -1 10 ", so that" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 1 "[" }{TEXT 277 1 "A" }{TEXT -1 1 "]" }{TEXT 373 3 " + \+ " }{TEXT -1 1 "[" }{TEXT 278 1 "B" }{TEXT -1 1 "]" }{TEXT 374 3 " = " }{TEXT -1 1 "[" }{TEXT 279 1 "B" }{TEXT -1 1 "]" }{TEXT 375 3 " + " } {TEXT -1 1 "[" }{TEXT 280 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 329 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 326 4 "N.B." }{TEXT -1 22 " Matrix addition is " }{TEXT 327 11 "asso ciative" }{TEXT -1 10 ", so that" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 1 "[" }{TEXT 281 1 "A" }{TEXT -1 1 "]" }{TEXT 376 4 " + (" }{TEXT -1 1 "[" }{TEXT 282 1 "B" }{TEXT -1 1 "]" }{TEXT 377 3 " + " }{TEXT -1 1 "[" }{TEXT 283 1 "C" }{TEXT -1 1 "]" }{TEXT 378 5 ") = (" }{TEXT -1 1 "[" }{TEXT 284 1 "A" }{TEXT -1 1 "]" }{TEXT 379 3 " + " }{TEXT -1 1 "[" }{TEXT 285 1 "B" }{TEXT -1 1 "]" }{TEXT 380 4 ") + " }{TEXT -1 1 "[" }{TEXT 286 1 "C" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT 311 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 4 "N.B." }{TEXT -1 17 " Two matrices, [" }{TEXT 314 1 "A" }{TEXT -1 7 "] and [" }{TEXT 315 1 "B" }{TEXT -1 18 "], with elements " } {XPPEDIT 18 0 "a[ij]" "6#&%\"aG6#%#ijG" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "b[ij]" "6#&%\"bG6#%#ijG" }{TEXT -1 23 " , respectively , are " }{TEXT 316 5 "equal" }{TEXT -1 44 " when they are both of th e same order and " }{XPPEDIT 18 0 "a[ij]=b[ij]" "6#/&%\"aG6#%#ijG&%\" bG6#F'" }{TEXT -1 7 " for " }{TEXT 320 3 "all" }{TEXT -1 29 " possi ble pairs of indices " }{TEXT 381 2 "( " }{XPPEDIT 18 0 "i" "6#%\"iG " }{TEXT 382 2 ", " }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT 383 2 " )" } {TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "[ Refer to Unit (10) where the function \+ " }{TEXT 470 5 "equal" }{TEXT -1 18 " contained in the " }{TEXT 469 6 "linalg" }{TEXT -1 38 " package is used for the first time. ]" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 317 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 36 "Numerical example of matrix addition" }} {PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {TEXT 384 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 385 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 386 1 ")" }{TEXT -1 12 " matrices \+ [" }{TEXT 287 1 "A" }{TEXT -1 7 "] and [" }{TEXT 288 1 "B" }{TEXT -1 13 "] be given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "A := m atrix(2, 3, [1, -2, 0, 3, 2, 1]) : B := matrix(2, 3, [-1, 3, 1, -2, \+ -1, 0]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) \+ ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrix G6#7$7%\"\"\"!\"#\"\"!7%\"\"$\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"BG-%'matrixG6#7$7%!\"\"\"\"$\"\"\"7%!\"#F*\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "T he sum [" }{TEXT 289 1 "A" }{TEXT -1 1 "]" }{TEXT 387 3 " + " }{TEXT -1 1 "[" }{TEXT 290 1 "B" }{TEXT -1 21 "] is the following " }{TEXT 388 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 389 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 390 1 ")" }{TEXT -1 9 " matrix:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`A+B` := matadd(A, B) : \+ A + B = matrix(`A+B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\" \"\"%\"BGF&-%'matrixG6#7$7%\"\"!F&F&7%F&F&F&" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-i n-a-book\" form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "matrix( A) + matrix(B) = matrix(`A+B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, &-%'matrixG6#7$7%\"\"\"!\"#\"\"!7%\"\"$\"\"#F*F*-F&6#7$7%!\"\"F.F*7%F+ F4F,F*-F&6#7$7%F,F*F*7%F*F*F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 270 "" 0 "" {TEXT 330 5 "* * *" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 331 4 "N.B." } {TEXT -1 2 " " }{TEXT 332 5 "Maple" }{TEXT -1 46 " recognises additio n of a number (scalar) and " }{TEXT 335 3 "any" }{TEXT -1 8 " matrix. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "For example, consider a square " }{TEXT 391 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 392 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 393 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 333 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A := matrix(3, 3, [a[11], a[12], a[13], a[21], a[22], a[23], a[31], a [32], a[33]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"AG-%'matrixG6#7%7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"# A&F+6#\"#B7%&F+6#\"#J&F+6#\"#K&F+6#\"#L" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "and the number \+ " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "The sum " } {XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT 394 3 " + " }{TEXT -1 1 "[" }{TEXT 334 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 395 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 396 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 397 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`k+A` := evalm(k + A) : k + A = matrix(`k+A `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"kG\"\"\"%\"AGF&-%'matrix G6#7%7%,&&%\"aG6#\"#6F&F%F&&F/6#\"#7&F/6#\"#87%&F/6#\"#@,&&F/6#\"#AF&F %F&&F/6#\"#B7%&F/6#\"#J&F/6#\"#K,&&F/6#\"#LF&F%F&" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Notice tha t the " }{TEXT 342 6 "matadd" }{TEXT -1 10 " function " }{TEXT 343 6 " cannot" }{TEXT -1 28 " be used to obtain the sum " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 1 " " }{TEXT 344 2 "+ " }{TEXT -1 1 "[" }{TEXT 345 1 "A" }{TEXT -1 36 "] since the function \"expects\" two " } {TEXT 398 8 "matrices" }{TEXT -1 23 " to be added together." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "It can be easily noticed that the result of this operation is e quivalent to the addition of a " }{TEXT 336 13 "scalar matrix" } {TEXT -1 3 " [" }{TEXT 337 1 "K" }{TEXT -1 15 "] of the same " } {TEXT 399 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 400 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 401 1 ")" }{TEXT -1 13 " order, vi z." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "K := diag(k, k, k) : K = matrix(K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"KG-%'matrixG6 #7%7%%\"kG\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "since the sum matrix [" } {TEXT 339 1 "K" }{TEXT -1 1 "]" }{TEXT 402 3 " + " }{TEXT -1 1 "[" } {TEXT 338 1 "A" }{TEXT -1 21 "] is the following " }{TEXT 403 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 404 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 405 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "`K+A` := evalm(K + A) : K + A = matrix(`K+ A`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"KG\"\"\"%\"AGF&-%'matri xG6#7%7%,&&%\"aG6#\"#6F&%\"kGF&&F/6#\"#7&F/6#\"#87%&F/6#\"#@,&&F/6#\"# AF&F2F&&F/6#\"#B7%&F/6#\"#J&F/6#\"#K,&&F/6#\"#LF&F2F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "[ Refer to Unit (7) for the " }{TEXT 340 13 "scalar matrix" }{TEXT -1 31 ", which is defined in Section " }{TEXT 445 1 "A" }{TEXT -1 24 ": \+ The diagonal matrix. ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 282 "" 0 "" {TEXT 434 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 435 4 "N.B." }{TEXT -1 39 " If addition of a number (scalar) to " }{TEXT 436 4 "each" } {TEXT -1 68 " element of a matrix is required, the simplest method is using the " }{TEXT 444 3 "map" }{TEXT -1 60 " function together with \+ the arrow-type functional operator " }{TEXT 438 2 "( " }{TEXT 437 3 " x->" }{TEXT 439 3 " )." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "For example, add the scalar " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 38 " to each element of the sa me square " }{TEXT 441 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 442 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 443 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 440 1 "A" }{TEXT -1 12 "] as before:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "`elements(A)+mu` := map(x->x+mu, A) : elements(A)+mu = matrix(`elements(A)+mu`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%)elementsG6#%\"AG\"\"\"%#muGF)-%'matrixG6#7%7%,&&% \"aG6#\"#6F)F*F),&&F26#\"#7F)F*F),&&F26#\"#8F)F*F)7%,&&F26#\"#@F)F*F), &&F26#\"#AF)F*F),&&F26#\"#BF)F*F)7%,&&F26#\"#JF)F*F),&&F26#\"#KF)F*F), &&F26#\"#LF)F*F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT 341 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT 319 1 "B" }{TEXT -1 2 ". " } {TEXT 275 18 "Matrix subtraction" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The difference [" }{TEXT 266 1 "A" }{TEXT -1 1 "]" }{TEXT 406 3 " \226 " }{TEXT -1 1 "[" } {TEXT 267 1 "B" }{TEXT -1 29 "] is defined by the relation" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 268 1 "A" }{TEXT -1 1 " ]" }{TEXT 407 3 " \226 " }{TEXT -1 1 "[" }{TEXT 269 1 "B" }{TEXT -1 1 "]" }{TEXT 408 3 " = " }{TEXT -1 1 "[" }{TEXT 271 1 "A" }{TEXT -1 1 "] " }{TEXT 409 4 " + (" }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT 410 1 ")" }{TEXT -1 1 "[" }{TEXT 270 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 " Therefore, similar rules and computation methods apply to matrix subtr action. This is illustrated hereunder for the matrices [" }{TEXT 272 1 "A" }{TEXT -1 7 "] and [" }{TEXT 273 1 "B" }{TEXT -1 50 "] containin g the same symbolic elements as before." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A := matrix (2, 3, [a[11], a[12], a[13], a[21], a[22], a[23]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B := matrix(2, 3, [b[11], b[12], b[13], b [21], b[22], b[23]]) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The difference [" }{TEXT 297 1 "A " }{TEXT -1 1 "]" }{TEXT 411 3 " \226 " }{TEXT -1 1 "[" }{TEXT 298 1 " B" }{TEXT -1 9 "] is a " }{TEXT 412 1 "(" }{XPPEDIT 18 0 "2" "6#\"\" #" }{TEXT 413 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 414 1 ") " }{TEXT -1 82 " matrix, which may be obtained using either of the fo llowing alternative methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 352 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 353 6 "matadd" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`A-B` := matadd(A, -B) : A - B = \+ matrix(`A-B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"%\"BG !\"\"-%'matrixG6#7$7%,&&%\"aG6#\"#6F&&%\"bGF1F(,&&F06#\"#7F&&F4F7F(,&& F06#\"#8F&&F4F " 0 "" {MPLTEXT 1 0 49 "`A-B` := evalm(A - B) : A - B = matrix(`A-B`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"%\"BG!\"\"-%'matrixG6#7$ 7%,&&%\"aG6#\"#6F&&%\"bGF1F(,&&F06#\"#7F&&F4F7F(,&&F06#\"#8F&&F4F " 0 "" {MPLTEXT 1 0 39 "matrix(A) - matrix(B) = matrix(`A-B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%' matrixG6#7$7%&%\"aG6#\"#6&F+6#\"#7&F+6#\"#87%&F+6#\"#@&F+6#\"#A&F+6#\" #B\"\"\"-F&6#7$7%&%\"bGF,&FDF/&FDF27%&FDF6&FDF9&FDFFCFK,&F.F>FEFK,&F1F>FFFK7%,&F5F>FHFK,&F8F>FIFK,&F;F>FJFK" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 39 " Numerical example of matrix subtraction" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Use the matrices \+ [" }{TEXT 291 1 "A" }{TEXT -1 7 "] and [" }{TEXT 292 1 "B" }{TEXT -1 52 "] with the same numerical elements as before, namely" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "A := matrix(2, 3, [1, -2, 0, 3, 2, \+ 1]) : B := matrix(2, 3, [-1, 3, 1, -2, -1, 0]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A = matrix(A) ; B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7$7%\"\"\"!\"#\"\"!7%\"\"$ \"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7%!\"\" \"\"$\"\"\"7%!\"#F*\"\"!" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The difference [" }{TEXT 295 1 "A " }{TEXT -1 1 "]" }{TEXT 415 3 " \226 " }{TEXT -1 1 "[" }{TEXT 296 1 " B" }{TEXT -1 21 "] is the following " }{TEXT 416 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 417 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" } {TEXT 418 1 ")" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`A-B` := matadd(A, -B) : A - B = matrix(`A-B`) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"AG\"\"\"%\"BG!\"\"-%'matrixG6#7 $7%\"\"#!\"&F(7%\"\"&\"\"$F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "matrix(A) - matrix(B) = matrix(`A-B`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%'matrixG6#7$7 %\"\"\"!\"#\"\"!7%\"\"$\"\"#F*F*-F&6#7$7%!\"\"F.F*7%F+F4F,F4-F&6#7$7%F /!\"&F47%\"\"&F.F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 265 "" 0 "" {TEXT 312 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 4 "N.B." }{TEXT -1 48 " S ubtraction of a matrix from itself yields a " }{TEXT 473 4 "zero" } {TEXT -1 6 " or " }{TEXT 474 4 "null" }{TEXT -1 16 " matrix whose \+ " }{TEXT 306 3 "all" }{TEXT -1 16 " elements are " }{XPPMATH 20 "6#% &zerosG" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 1 "[" }{TEXT 299 1 "A" } {TEXT -1 1 "]" }{TEXT 419 3 " \226 " }{TEXT -1 1 "[" }{TEXT 300 1 "A" }{TEXT -1 1 "]" }{TEXT 420 3 " = " }{TEXT -1 1 "[" }{TEXT 301 1 "0" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "To illustrate this case, use the above ma trix [" }{TEXT 303 1 "A" }{TEXT -1 48 "] with its numerical elements. \+ The difference [" }{TEXT 304 1 "A" }{TEXT -1 1 "]" }{TEXT 421 3 " \+ \226 " }{TEXT -1 1 "[" }{TEXT 305 1 "A" }{TEXT -1 21 "] is the follow ing " }{TEXT 422 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 423 3 " \+ \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 424 1 ")" }{TEXT -1 2 " " }{TEXT 475 4 "zero" }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "`A-A` := matadd(A, -A) : `A \226 A` = matrix(`A-A `) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&A~|at~AG-%'matrixG6#7$7%\" \"!F*F*F)" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "or, in \"like-in-a-book\" form," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "[0] = matrix(`A-A`) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/7#\"\"!-%'matrixG6#7$7%F%F%F%F*" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Notice tha t the " }{TEXT 432 5 "evalm" }{TEXT -1 93 " function cannot be used fo r performing this operation because it returns the scalar number " } {XPPMATH 20 "6#%%zeroG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`A-A` := evalm(A - A ) : `A \226 A` = `A-A` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&A~|at~AG\"\"!" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 281 "" 0 "" {TEXT 433 5 "* * *" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (4) for \"" }{TEXT 431 26 "Multiplication of matrices" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 262 "" 0 "" {TEXT 430 67 "------------------------------------------------------------------ -" }}}}{MARK "5 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }