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1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 128 0 128 1 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 "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier " 1 12 0 128 128 1 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 "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 }{PSTYLE "Normal" -1 260 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 363 38 "MATRICES AND MATRIX OPE RATIONS: Unit 2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 365 23 "Dr. Wlodzislaw Kostecki" }}{PARA 260 " " 0 "" {TEXT -1 53 "The Papua New Guinea University of Technology (PNG UT)" }}{PARA 260 "" 0 "" {TEXT -1 54 "Department of Electrical and Com munication Engineering" }}{PARA 260 "" 0 "" {TEXT -1 20 "Lae, Morobe P rovince" }}{PARA 260 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 364 41 "Copyright \251 2000 by Wlodzislaw Kostecki" }}{PARA 258 "" 0 "" {TEXT 366 19 "All \+ rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 367 67 "-------------------------------------------------------- -----------" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 3 "(2)" }{TEXT 259 1 " " }{TEXT 258 33 "Extracting components of matrices" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 376 10 "OBJECTIVES" }{TEXT 377 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 378 1 "\225" }{TEXT -1 101 " To provide alternative methods of extrac ting individual elements and the various parts of a matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 381 1 "\225" }{TEXT -1 34 " To introduce the concepts of a " }{TEXT 382 8 "sequence" }{TEXT -1 7 " and " } {TEXT 383 3 "set" }{TEXT -1 6 " of " }{TEXT 384 8 "elements" }{TEXT -1 10 " and of " }{TEXT 385 8 "sequence" }{TEXT -1 6 " of " } {TEXT 386 5 "lists" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 387 1 "\225" }{TEXT -1 31 " To introduce the concept of " }{TEXT 388 18 "subscript notation" }{TEXT -1 27 " and show its application. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 395 1 "\225" }{TEXT -1 32 " To show how to convert (row) " }{TEXT 396 7 "vectors" }{TEXT -1 6 " to " } {TEXT 397 15 "column matrices" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 379 1 "\225" }{TEXT -1 96 " To show how to create new matric es using extracted parts of rows or complete rows of a matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 380 1 "\225" }{TEXT -1 102 " To show how to create new matrices using extracted parts of columns or complete c olumns of a matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "restart : with(linalg, au gment, col, coldim, concat, delcols, delrows, row, rowdim, stackmatrix , submatrix, subvector, transpose) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 548 "There are many situati ons where the various components of a matrix are needed for further ma thematical treatment. This may include extracting all matrix elements, but most often desired are specified individual elements, parts of ro ws or columns, entire rows or columns, a number of consecutive rows or columns, and submatrices. All of them may be extracted from a given m atrix using suitable functions and operations. Extracting components o f matrices is also necessary in some other Units. An overview of the p ertinent methods is given hereunder." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 336 1 "(" }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 337 3 " \327 \+ " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 338 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 262 1 "A" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "A := matrix(4, 4, [a[11], a[12], a[13], a[14], \+ a[21], a[22], a[23], a[24], a[31], a[32], a[33], a[34], a[41], a[42], \+ a[43], a[44]]) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"AG-%'matrixG6#7&7&&%\"aG6#\"#6&F+6#\"#7&F+6#\"#8&F+6#\"#97&&F+6#\" #@&F+6#\"#A&F+6#\"#B&F+6#\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M7&& F+6#\"#T&F+6#\"#U&F+6#\"#V&F+6#\"#W" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 261 1 "A" }{TEXT -1 2 ". " } {TEXT 260 34 "Extracting all entries of a matrix" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 8 "Method 1" }{TEXT -1 46 ". Returning matrix elements in the form of a " }{TEXT 270 8 "sequence" }{TEXT -1 29 " in which the elements are " }{TEXT 399 7 "ordered" }{TEXT -1 13 " row-by-row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "elements(A) := op(1..rowdim(A)*coldim(A), convert( A, set)) : elements(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "62&%\"aG6# \"#6&F$6#\"#7&F$6#\"#8&F$6#\"#9&F$6#\"#@&F$6#\"#A&F$6#\"#B&F$6#\"#C&F$ 6#\"#J&F$6#\"#K&F$6#\"#L&F$6#\"#M&F$6#\"#T&F$6#\"#U&F$6#\"#V&F$6#\"#W " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 8 "Method 2" }{TEXT -1 52 ". Returning matrix elements in th e form of a (row) " }{TEXT 264 6 "vector" }{TEXT -1 41 " in which th e elements are enclosed in " }{TEXT 400 8 "brackets" }{TEXT -1 7 " a nd " }{TEXT 401 7 "ordered" }{TEXT -1 13 " row-by-row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "elements(A) := convert(A, vector) \+ : elements(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#72&%\" aG6#\"#6&F(6#\"#7&F(6#\"#8&F(6#\"#9&F(6#\"#@&F(6#\"#A&F(6#\"#B&F(6#\"# C&F(6#\"#J&F(6#\"#K&F(6#\"#L&F(6#\"#M&F(6#\"#T&F(6#\"#U&F(6#\"#V&F(6# \"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 52 "[ Refer to Unit (5) for more information about the " } {TEXT 389 6 "vector" }{TEXT -1 3 ". ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 8 "Method 3" }{TEXT -1 46 ". Returning matrix elements in the form of a " }{TEXT 273 3 "s et" }{TEXT -1 41 " in which the elements are enclosed in " }{TEXT 398 6 "braces" }{TEXT -1 7 " and " }{TEXT 402 7 "ordered" }{TEXT -1 13 " row-by-row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "elemen ts(A) := convert(A, set) : elements(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<2&%\"aG6#\"#6&F%6#\"#7&F%6#\"#8&F%6#\"#9&F%6#\"#@&F%6# \"#A&F%6#\"#B&F%6#\"#C&F%6#\"#J&F%6#\"#K&F%6#\"#L&F%6#\"#M&F%6#\"#T&F% 6#\"#U&F%6#\"#V&F%6#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 274 8 "Method 4" }{TEXT -1 46 ". Returnin g matrix elements in the form of a " }{TEXT 275 4 "list" }{TEXT -1 41 " in which the elements are enclosed in " }{TEXT 403 8 "brackets " }{TEXT -1 7 " and " }{TEXT 404 7 "ordered" }{TEXT -1 13 " row-by- row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "elements(A) := conv ert(convert(A, set), list) : elements(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72&%\"aG6#\"#6&F%6#\"#7&F%6#\"#8&F%6#\"#9&F%6#\"#@&F%6# \"#A&F%6#\"#B&F%6#\"#C&F%6#\"#J&F%6#\"#K&F%6#\"#L&F%6#\"#M&F%6#\"#T&F% 6#\"#U&F%6#\"#V&F%6#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 265 8 "Method 5" }{TEXT -1 46 ". Returnin g matrix elements in the form of a " }{TEXT 266 13 "list of lists" } {TEXT -1 85 " in which each list comprises the elements of one row, t he lists being enclosed in " }{TEXT 405 8 "brackets" }{TEXT -1 7 " a nd " }{TEXT 406 7 "ordered" }{TEXT -1 13 " row-by-row:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "elements(A) := convert(A, listlist) : elements(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7&&%\"aG6#\"#6 &F&6#\"#7&F&6#\"#8&F&6#\"#97&&F&6#\"#@&F&6#\"#A&F&6#\"#B&F&6#\"#C7&&F& 6#\"#J&F&6#\"#K&F&6#\"#L&F&6#\"#M7&&F&6#\"#T&F&6#\"#U&F&6#\"#V&F&6#\"# W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 8 "Method 6" }{TEXT -1 46 ". Returning matrix elements in th e form of an " }{TEXT 269 9 "unordered" }{TEXT -1 2 " " }{TEXT 268 17 "sequence of lists" }{TEXT -1 52 " in which each list contains one bracketed element:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "elem ents(A) := entries(A) : elements(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "627#&%\"aG6#\"#U7#&F%6#\"#W7#&F%6#\"#67#&F%6#\"#L7#&F%6#\"#@7#&F%6 #\"#V7#&F%6#\"#97#&F%6#\"#77#&F%6#\"#A7#&F%6#\"#C7#&F%6#\"#J7#&F%6#\"# 87#&F%6#\"#T7#&F%6#\"#B7#&F%6#\"#M7#&F%6#\"#K" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "B. " }{TEXT 276 41 "Extracting individual entries of a matrix" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "To extract specified entries of a matrix, use the " }{TEXT 305 18 "subscript no tation" }{TEXT -1 34 " indicating the element location." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "F or example, extract the elements at locations " }{TEXT 279 5 "(1,1)" }{TEXT -1 7 " and " }{TEXT 280 5 "(2,3)" }{TEXT -1 13 " of matrix [ " }{TEXT 278 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "el[11] := A[1,1] : el[23] := A[2,3] : element[11] (A) = el[11] ; element[23](A) = el[23] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%(elementG6#\"#66#%\"AG&%\"aGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%(elementG6#\"#B6#%\"AG&%\"aGF'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "C. " } {TEXT 321 36 "Extracting part of a row of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "To extrac t specified part of a row of a matrix, use the " }{TEXT 322 9 "subvect or" }{TEXT -1 118 " function with a number indicating the row location , followed by number range indicating locations of desired columns." } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "For example, extract elements " }{XPPEDIT 18 0 "1" "6#\" \"\"" }{TEXT 324 3 "..." }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 10 " \+ of row " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 17 " and elements \+ " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 325 3 "..." }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT -1 10 " of row " }{XPPEDIT 18 0 "4" "6#\"\"%" } {TEXT -1 13 " of matrix [" }{TEXT 323 1 "A" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Row[2][1..3](A) := subvector (A, 2, 1..3) : Row[4][2..3](A) := subvector(A, 4, 2..3) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "'Row[2][1..3](A)' = Row[2][1 ..3](A) ; 'Row[4][2..3](A)' = Row[4][2..3](A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%$RowG6#\"\"#6#;\"\"\"\"\"$6#%\"AG-%'vectorG6#7%&% \"aG6#\"#@&F56#\"#A&F56#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%$ RowG6#\"\"%6#;\"\"#\"\"$6#%\"AG-%'vectorG6#7$&%\"aG6#\"#U&F56#\"#V" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "To create the corresponding " }{TEXT 327 12 "row matrices" } {TEXT -1 70 ", each consisting of elements of parts of the selected r ows, use the " }{TEXT 326 6 "matrix" }{TEXT -1 10 " function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "RM[2][1..3] := matrix(1, 3, \+ [Row[2][1..3](A)]) : RM[4][2..3] := matrix(1, 2, [Row[4][2..3](A)]) \+ :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "RM[Row[2]][1..3](A) = \+ matrix(RM[2][1..3]) ; RM[Row[4]][2..3](A) = matrix(RM[4][2..3]) ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%#RMG6#&%$RowG6#\"\"#6#;\"\"\"\" \"$6#%\"AG-%'matrixG6#7#7%&%\"aG6#\"#@&F96#\"#A&F96#\"#B" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-&&%#RMG6#&%$RowG6#\"\"%6#;\"\"#\"\"$6#%\"AG-%' matrixG6#7#7$&%\"aG6#\"#U&F96#\"#V" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "D. " }{TEXT 328 39 "Ex tracting part of a column of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To extract specif ied part of a column of a matrix, use the " }{TEXT 329 9 "subvector" } {TEXT -1 120 " function with a number range indicating locations of de sired rows, followed by a number indicating the column location." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "For example, extract elements " }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 332 3 "..." }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 13 " of colu mn " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 17 " and elements " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 331 3 "..." }{XPPEDIT 18 0 "3" "6# \"\"$" }{TEXT -1 13 " of column " }{XPPEDIT 18 0 "4" "6#\"\"%" } {TEXT -1 13 " of matrix [" }{TEXT 330 1 "A" }{TEXT -1 2 "]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Col[2][2..3](A) := subvector (A, 2..3, 1) : Col[4][1..3](A) := subvector(A, 1..3, 4) :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "'Col[2][2..3](A)' = Col[2][2 ..3](A) ; 'Col[4][1..3](A)' = Col[4][1..3](A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%$ColG6#\"\"#6#;F)\"\"$6#%\"AG-%'vectorG6#7$&%\"aG6 #\"#@&F46#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%$ColG6#\"\"%6#; \"\"\"\"\"$6#%\"AG-%'vectorG6#7%&%\"aG6#\"#9&F56#\"#C&F56#\"#M" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "To convert the specified parts of columns into " }{TEXT 335 15 "column matrices" }{TEXT -1 75 ", each consisting of elements of t he selected part of the column, use the " }{TEXT 333 7 "convert" } {TEXT -1 5 " and " }{TEXT 334 6 "matrix" }{TEXT -1 11 " functions:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "CM[2][2..3] := convert(Col [2][2..3](A), matrix) : CM[4][1..3] := convert(Col[4][1..3](A), matr ix) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "CM[Col[2]][2..3](A ) = matrix(CM[2][2..3]) ; CM[Col[4]][1..3](A) = matrix(CM[4][1..3]) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%#CMG6#&%$ColG6#\"\"#6#;F,\" \"$6#%\"AG-%'matrixG6#7$7#&%\"aG6#\"#@7#&F86#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&&%#CMG6#&%$ColG6#\"\"%6#;\"\"\"\"\"$6#%\"AG-%'matrix G6#7%7#&%\"aG6#\"#97#&F96#\"#C7#&F96#\"#M" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "E. " }{TEXT 277 38 "Extracting individual rows of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "To extract specif ied rows of a matrix, use the " }{TEXT 281 3 "row" }{TEXT -1 76 " func tion with a number indicating the row location. The output is a (row) \+ " }{TEXT 282 6 "vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "For example, extr act rows " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 13 " of matrix [" }{TEXT 283 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Row [1](A):=row(A, 1) : Row[3](A):=row(A, 3) : 'Row[1](A)' = Row[1](A) \+ ; 'Row[3](A)' = Row[3](A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$R owG6#\"\"\"6#%\"AG-%'vectorG6#7&&%\"aG6#\"#6&F06#\"#7&F06#\"#8&F06#\"# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$RowG6#\"\"$6#%\"AG-%'vectorG 6#7&&%\"aG6#\"#J&F06#\"#K&F06#\"#L&F06#\"#M" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "To create the c orresponding " }{TEXT 295 12 "row matrices" }{TEXT -1 60 ", each con sisting of elements of the selected row, use the " }{TEXT 294 6 "matri x" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "RM[1] := matrix(1, 4, [Row[1](A)]) : RM[3] := matrix(1, 4, [Row[ 3](A)]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "RM[Row[1]](A) \+ = matrix(RM[1]) ; RM[Row[3]](A) = matrix(RM[3]) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-&%#RMG6#&%$RowG6#\"\"\"6#%\"AG-%'matrixG6#7#7&&%\"a G6#\"#6&F46#\"#7&F46#\"#8&F46#\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-&%#RMG6#&%$RowG6#\"\"$6#%\"AG-%'matrixG6#7#7&&%\"aG6#\"#J&F46#\"#K&F 46#\"#L&F46#\"#M" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "F. " }{TEXT 284 41 "Extracting individua l columns of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "To extract specified columns of a \+ matrix, use the " }{TEXT 285 3 "col" }{TEXT -1 79 " function with a nu mber indicating the column location. The output is a (row) " }{TEXT 286 6 "vector" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "For example, extract columns \+ " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 13 " of matrix [" }{TEXT 287 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "Col[1](A) := col( A, 1) : Col[3](A) := col(A, 3) : 'Col[1](A)' = Col[1](A) ; 'Col[ 3](A)' = Col[3](A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$ColG6#\" \"\"6#%\"AG-%'vectorG6#7&&%\"aG6#\"#6&F06#\"#@&F06#\"#J&F06#\"#T" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%$ColG6#\"\"$6#%\"AG-%'vectorG6#7&& %\"aG6#\"#8&F06#\"#B&F06#\"#L&F06#\"#V" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "To convert the sp ecified columns into " }{TEXT 296 15 "column matrices" }{TEXT -1 63 " , each consisting of elements of the selected column, use the " } {TEXT 288 7 "convert" }{TEXT -1 5 " and " }{TEXT 289 6 "matrix" } {TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "CM[1] := convert(Col[1](A), matrix) : CM[3] := convert(Col[3](A), m atrix) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "CM[Col[1]](A) = matrix(CM[1]) ; CM[Col[3]](A) = matrix(CM[3]) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-&%#CMG6#&%$ColG6#\"\"\"6#%\"AG-%'matrixG6#7&7#&%\"a G6#\"#67#&F46#\"#@7#&F46#\"#J7#&F46#\"#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%#CMG6#&%$ColG6#\"\"$6#%\"AG-%'matrixG6#7&7#&%\"aG6#\"#87#&F4 6#\"#B7#&F46#\"#L7#&F46#\"#V" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "G. " }{TEXT 290 47 "Extracting several consecutive rows of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "To extract severa l consecutive rows of a matrix, use the " }{TEXT 291 3 "row" }{TEXT -1 95 " function with a number range indicating the row locations. The output is a sequence of (row) " }{TEXT 292 7 "vectors" }{TEXT -1 1 " ." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "For example, extract rows " }{XPPEDIT 18 0 "2" "6#\"\"# " }{TEXT 373 3 "..." }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT -1 13 " of m atrix [" }{TEXT 293 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "Rows[2..3](A) := row(A, 2..3) : 'Rows[2..3](A)' = Rows[2..3](A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%%RowsG6#;\"\"# \"\"$6#%\"AG6$-%'vectorG6#7&&%\"aG6#\"#@&F36#\"#A&F36#\"#B&F36#\"#C-F/ 6#7&&F36#\"#J&F36#\"#K&F36#\"#L&F36#\"#M" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "To create a matri x [" }{TEXT 358 1 "B" }{TEXT -1 71 "] consisting of the selected rows, use either of the following methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 359 8 "Method 1" }{TEXT -1 62 ". Using conversion of (row) vectors into row matrices and the \+ " }{TEXT 360 11 "stackmatrix" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 340 6 "S tep 1" }{TEXT -1 154 ". Extract individual vectors from the above sequ ence of vectors using the subscript notation and then, convert either \+ of them into a row matrix using the " }{TEXT 339 6 "matrix" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "RM[Row[2] ] := matrix(1, 4, [Rows[2..3](A)[1]]) : RM[Row[3]] := matrix(1, 4, [ Rows[2..3](A)[2]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "'RM [Row[2]]'(A) = matrix(RM[Row[2]]) ; 'RM[Row[3]]'(A) = matrix(RM[Row[ 3]]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%#RMG6#&%$RowG6#\"\"#6#% \"AG-%'matrixG6#7#7&&%\"aG6#\"#@&F46#\"#A&F46#\"#B&F46#\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%#RMG6#&%$RowG6#\"\"$6#%\"AG-%'matrixG6#7 #7&&%\"aG6#\"#J&F46#\"#K&F46#\"#L&F46#\"#M" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 343 6 "Step 2" }{TEXT -1 49 ". Use the row matrix with the highest row index (" }{TEXT 341 10 "RM[Row[3]]" }{TEXT -1 81 " in this case) as the reference and stac k a row matrix with the lower row index (" }{TEXT 342 10 "RM[Row[2]]" }{TEXT -1 72 " in this case) directly on top of the reference matrix b y employing the " }{TEXT 344 5 "stack" }{TEXT -1 10 " function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "B := stackmatrix(RM[Row[2]], RM[Row[3]]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"BG-%'matrixG6#7$7&&%\"aG6#\"#@&F+6#\"#A&F+6#\"#B&F+6#\"#C7&&F+6#\"#J &F+6#\"#K&F+6#\"#L&F+6#\"#M" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 393 4 "N.B." }{TEXT -1 15 " The functi on " }{TEXT 394 11 "stackmatrix" }{TEXT -1 50 " joins two or more matr ices or vectors vertically." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 362 8 "Method 2" }{TEXT -1 114 ". Using (row) vectors extracted from the above sequence of vectors, typed in \+ a proper order in brackets under the " }{TEXT 361 6 "matrix" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "B := m atrix(2, 4, [Rows[2..3](A)[1], Rows[2..3](A)[2]]) : B= matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7$7&&%\"aG6#\"#@&F +6#\"#A&F+6#\"#B&F+6#\"#C7&&F+6#\"#J&F+6#\"#K&F+6#\"#L&F+6#\"#M" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "H. " }{TEXT 297 50 "Extracting several consecutive columns of a matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "To extract several consecutive columns of a mat rix, use the " }{TEXT 298 3 "col" }{TEXT -1 98 " function with a numbe r range indicating the column locations. The output is a sequence of ( row) " }{TEXT 299 7 "vectors" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "For example, e xtract columns " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 372 3 "..." } {XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 13 " of matrix [" }{TEXT 300 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Cols [3..4](A) := col(A, 3..4) : 'Cols[3..4](A)' = Cols[3..4](A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%%ColsG6#;\"\"$\"\"%6#%\"AG6$-%'vec torG6#7&&%\"aG6#\"#8&F36#\"#B&F36#\"#L&F36#\"#V-F/6#7&&F36#\"#9&F36#\" #C&F36#\"#M&F36#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "To create a matrix [" }{TEXT 301 1 "C" }{TEXT -1 74 "] consisting of the selected columns, use either o f the following methods." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 345 8 "Method 1" }{TEXT -1 65 ". Using co nversion of (row) vectors into column matrices and the " }{TEXT 346 7 "augment" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 304 6 "Step 1" }{TEXT -1 157 " . Extract individual vectors from the above sequence of vectors using \+ the subscript notation and then, convert either of them into a column \+ matrix using the " }{TEXT 302 7 "convert" }{TEXT -1 5 " and " }{TEXT 303 6 "matrix" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "CM[Col[3]] := convert(Cols[3..4](A)[1], matrix) : \+ CM[Col[4]] := convert(Cols[3..4](A)[2], matrix) :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 79 "'CM[Col[3]]'(A) = matrix(CM[Col[3]]) ; 'CM [Col[4]]'(A) = matrix(CM[Col[4]]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-&%#CMG6#&%$ColG6#\"\"$6#%\"AG-%'matrixG6#7&7#&%\"aG6#\"#87#&F46#\"# B7#&F46#\"#L7#&F46#\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%#CMG6#& %$ColG6#\"\"%6#%\"AG-%'matrixG6#7&7#&%\"aG6#\"#97#&F46#\"#C7#&F46#\"#M 7#&F46#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 308 6 "Step 2" }{TEXT -1 55 ". Use the column matrix w ith the highest column index (" }{TEXT 306 10 "CM[Col[4]]" }{TEXT -1 86 " in this case) as the reference and join a column matrix with the \+ lower column index (" }{TEXT 307 10 "CM[Col[3]]" }{TEXT -1 84 " in thi s case) directly to the left of the reference matrix by employing eith er the " }{TEXT 309 7 "augment" }{TEXT -1 4 " or " }{TEXT 310 6 "conca t" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "C := augment(CM[Col[3]], CM[Col[4]]) : C := concat(CM[Col[3]], C M[Col[4]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "C = matrix( C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7&7$&%\"aG6# \"#8&F+6#\"#97$&F+6#\"#B&F+6#\"#C7$&F+6#\"#L&F+6#\"#M7$&F+6#\"#V&F+6# \"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 392 4 "N.B." }{TEXT -1 15 " The function " }{TEXT 390 7 "augm ent" }{TEXT -1 16 " or its synonym " }{TEXT 391 6 "concat" }{TEXT -1 61 " joins two or more matrices or vectors together horizontally." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 347 8 "Method 2" }{TEXT -1 62 ". Using conversion of (row) vectors int o row matrices and the " }{TEXT 348 11 "stackmatrix" }{TEXT -1 5 " and " }{TEXT 349 9 "transpose" }{TEXT -1 11 " functions:" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 351 6 "S tep 1" }{TEXT -1 154 ". Extract individual vectors from the above sequ ence of vectors using the subscript notation and then, convert either \+ of them into a row matrix using the " }{TEXT 350 6 "matrix" }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "RM[Col[3] ] := matrix(1, 4, [Cols[3..4](A)[1]]) : RM[Col[4]] := matrix(1, 4, [ Cols[3..4](A)[2]]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "'RM [Col[3]]'(A) = matrix(RM[Col[3]]) ; 'RM[Col[4]]'(A) = matrix(RM[Col[ 4]]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%#RMG6#&%$ColG6#\"\"$6#% \"AG-%'matrixG6#7#7&&%\"aG6#\"#8&F46#\"#B&F46#\"#L&F46#\"#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%#RMG6#&%$ColG6#\"\"%6#%\"AG-%'matrixG6#7 #7&&%\"aG6#\"#9&F46#\"#C&F46#\"#M&F46#\"#W" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 354 6 "Step 2" }{TEXT -1 52 ". Use the row matrix with the highest column index (" }{TEXT 352 10 "RM[Col[4]]" }{TEXT -1 84 " in this case) as the reference and \+ stack a row matrix with the lower column index (" }{TEXT 353 10 "RM[Co l[3]]" }{TEXT -1 72 " in this case) directly on top of the reference m atrix by employing the " }{TEXT 355 11 "stackmatrix" }{TEXT -1 84 " fu nction. Then, interchange the rows and columns of the resultant matrix using the " }{TEXT 356 9 "transpose" }{TEXT -1 10 " function:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "C := transpose(stackmatrix(R M[Col[3]], RM[Col[4]])) : C = matrix(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG-%'matrixG6#7&7$&%\"aG6#\"#8&F+6#\"#97$&F+6#\"#B& F+6#\"#C7$&F+6#\"#L&F+6#\"#M7$&F+6#\"#V&F+6#\"#W" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the \+ " }{TEXT 357 9 "transpose" }{TEXT -1 36 " of a matrix, refer to Unit (10). ]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "I. " }{TEXT 311 34 "Extracting a submatrix of a ma trix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 104 "To extract a submatrix containing the elements specifi ed by the row range and the column range, use the " }{TEXT 312 9 "subm atrix" }{TEXT -1 112 " function with a number range indicating row loc ations, followed by a number range indicating column locations.." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "For example, extract a submatrix contained between rows " } {XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 374 3 "..." }{XPPEDIT 18 0 "3" "6# \"\"$" }{TEXT -1 15 " and columns " }{XPPEDIT 18 0 "2" "6#\"\"#" } {TEXT 375 3 "..." }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT -1 13 " of matr ix [" }{TEXT 370 1 "A" }{TEXT -1 2 "]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S := submatrix(A, 2..3, 2..4) : S = matrix(S) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"SG-%'matrixG6#7$7%&%\"aG6#\"#A&F+6 #\"#B&F+6#\"#C7%&F+6#\"#K&F+6#\"#L&F+6#\"#M" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 4 "N.B." }{TEXT -1 80 " The same result may be obtained using a suitable combination \+ of the functions " }{TEXT 319 7 "delcols" }{TEXT -1 8 ", which " } {TEXT 314 3 "del" }{TEXT -1 5 "etes " }{TEXT 315 3 "col" }{TEXT -1 3 " umn" }{TEXT 316 1 "s" }{TEXT -1 5 " and " }{TEXT 320 7 "delrows" } {TEXT -1 8 ", which " }{TEXT 317 3 "del" }{TEXT -1 5 "etes " }{TEXT 318 4 "rows" }{TEXT -1 67 " of a matrix. A combination appropriate for this particular case is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "S := delrows(delrows(delcols(A, 1..1), 1..1), 3..3) : S = matrix(S) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"SG-%'matrixG6#7$7%&%\"aG6#\" #A&F+6#\"#B&F+6#\"#C7%&F+6#\"#K&F+6#\"#L&F+6#\"#M" }}}{EXCHG {PARA 2 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 371 5 "* * *" } }}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Proceed to Unit (3) for \"" }{TEXT 369 36 "Addition and s ubtraction of matrices" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 258 "" 0 "" {TEXT 368 67 "-------------------------------------------------------- -----------" }}}}{MARK "156 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }