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{TEXT 359 19 "All rights reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT 360 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 263 3 "(9)" }{TEXT 318 1 " " }{TEXT 317 40 "Multiplication inv olving the unit matrix" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 371 10 "OBJECTIVES" }{TEXT 372 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 373 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 374 11 "unit matr ix" }{TEXT -1 6 " or " }{TEXT 375 15 "identity matrix" }{TEXT -1 1 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 376 1 "\225" }{TEXT -1 62 " To pro vide alternative methods of inputting the unit matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 377 1 "\225" }{TEXT -1 81 " To provide an examp le of multiplication of a square matrix and the unit matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 380 1 "\225" }{TEXT -1 68 " To provide a n example of multiplication of the unit matrix and a " }{TEXT 379 13 "column matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 381 1 "\225" }{TEXT -1 48 " To provide an example of multiplication of a \+ " }{TEXT 382 10 "row matrix" }{TEXT -1 22 " and the unit matrix." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 378 1 "\225" }{TEXT -1 63 " To specify a nd illustrate some properties of the unit matrix." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rest art : with(linalg, det, diag, multiply) :" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "A " }{TEXT 370 6 "scalar" }{TEXT -1 21 " matrix, in which " }{TEXT 265 3 "all" } {TEXT -1 29 " the diagonal elements are " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 17 " is called the " }{TEXT 266 11 "unit matrix" }{TEXT -1 6 " or " }{TEXT 267 15 "identity matrix" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "In \+ " }{TEXT 300 5 "Maple" }{TEXT -1 12 ", there are " }{TEXT 315 3 "two" }{TEXT -1 63 " alternative methods of defining and inputting the unit \+ matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 323 1 "(" }{XPPEDIT 18 0 "3" "6 #\"\"$" }{TEXT 324 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 325 1 ")" }{TEXT -1 15 " unit matrix [" }{TEXT 264 1 "U" }{TEXT -1 2 "]. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 319 8 "Method 1" }{TEXT -1 12 ". Using the " }{TEXT 320 4 "diag " }{TEXT -1 10 " function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "U := diag(1, 1, 1) : U = matrix(U) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 8 "Method 2" }{TEXT -1 12 ". Using the " }{TEXT 322 8 "identity" } {TEXT -1 19 " indexing function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "U := array(1..3, 1..3, identity) : U = matrix(U) ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"UG-%'matrixG6#7%7%\"\"\"\"\"!F +7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 305 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 4 "N.B." }{TEXT -1 19 " T he unit matrix " }{TEXT 268 8 "commutes" }{TEXT -1 34 " in multiplic ation, i.e. for any " }{TEXT 271 6 "square" }{TEXT -1 9 " matrix [" } {TEXT 270 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 1 "[" }{TEXT 272 1 "U" }{TEXT -1 3 "] [" }{TEXT 273 1 "A" }{TEXT 274 4 "] = " }{TEXT -1 1 "[" }{TEXT 275 1 "A" }{TEXT -1 3 "] [" }{TEXT 276 1 "U" }{TEXT 277 4 "] = " }{TEXT -1 1 "[" }{TEXT 278 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "where the order of [" }{TEXT 279 1 "U" }{TEXT -1 26 "] is the same as that of [" }{TEXT 280 1 "A" }{TEXT -1 2 "]." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 326 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 327 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 328 1 ")" }{TEXT -1 2 " " }{TEXT 286 11 "unit matrix" }{TEXT -1 3 " \+ [" }{TEXT 281 1 "U" }{TEXT -1 9 "] and a " }{TEXT 329 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 330 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 331 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 295 1 "A" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " A := matrix(3, 3, [8, 2, 1, 1, 3, 0, 4, 2, 1]) : A = matrix(A) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matrixG6#7%7%\"\")\"\"#\"\"\" 7%F,\"\"$\"\"!7%\"\"%F+F," }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(a) The product [" }{TEXT 282 1 " U" }{TEXT -1 3 "] [" }{TEXT 296 1 "A" }{TEXT -1 21 "] is the followin g " }{TEXT 332 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 333 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 334 1 ")" }{TEXT -1 9 " matrix :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`UA` := multiply(U, A) : U*A = matrix(`UA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"UG \"\"\"%\"AGF&-%'matrixG6#7%7%\"\")\"\"#F&7%F&\"\"$\"\"!7%\"\"%F.F&" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book\" form, viz ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "matrix(U) * matrix(A) \+ = matrix(`UA`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7%7 %\"\"\"\"\"!F+7%F+F*F+7%F+F+F*F*-F&6#7%7%\"\")\"\"#F*7%F*\"\"$F+7%\"\" %F3F*F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "restart : with(linalg, diag, multiply) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A := matrix(3, 3, [8, 2, \+ 1, 1, 3, 0, 4, 2, 1]) : U := diag(1, 1, 1) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The prod uct [" }{TEXT 297 1 "A" }{TEXT -1 3 "] [" }{TEXT 283 1 "U" }{TEXT -1 21 "] is the following " }{TEXT 335 1 "(" }{XPPEDIT 18 0 "3" "6#\"\" $" }{TEXT 336 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 337 1 ") " }{TEXT -1 9 " matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "`AU` := multiply(A, U) : A*U = matrix(`AU`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"AG\"\"\"%\"UGF&-%'matrixG6#7%7%\"\")\"\"#F&7%F&\" \"$\"\"!7%\"\"%F.F&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation may be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "matrix(A) * matrix(U) = matrix(`AU`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F,\"\"$\"\"!7%\"\" %F+F,F,-F&6#7%7%F,F/F/7%F/F,F/7%F/F/F,F,F%" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 306 5 "* * *" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 4 "N.B." }{TEXT -1 22 " The product of the " }{TEXT 257 11 "unit matrix" }{TEXT -1 9 " and a " }{TEXT 259 13 "column matrix" }{TEXT -1 10 " is the " }{TEXT 303 6 "matrix" }{TEXT -1 2 " " }{TEXT 304 9 "unchanged" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " } {TEXT 338 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 339 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 340 1 ")" }{TEXT -1 2 " " }{TEXT 287 11 "unit matrix" }{TEXT -1 3 " [" }{TEXT 284 1 "U" }{TEXT -1 9 "] and a " }{TEXT 341 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 342 3 " \327 " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 343 1 ")" }{TEXT -1 2 " \+ " }{TEXT 288 13 "column matrix" }{TEXT -1 3 " [" }{TEXT 285 2 "CM" } {TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 " CM := matrix(3, 1, [cm[11], cm[21], cm[31]]) : CM = matrix(CM) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#CMG-%'matrixG6#7%7#&%#cmG6#\"#67#&F +6#\"#@7#&F+6#\"#J" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The product [" }{TEXT 289 1 "U" }{TEXT -1 3 "] [" }{TEXT 290 2 "CM" }{TEXT -1 21 "] is the following " } {TEXT 344 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 345 3 " \327 " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 346 1 ")" }{TEXT -1 16 " column m atrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`U CM` := multipl y(U, CM) : U*CM = matrix(`U CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&%\"UG\"\"\"%#CMGF&-%'matrixG6#7%7#&%#cmG6#\"#67#&F.6#\"#@7#&F.6# \"#J" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 62 "This operation may be displayed in \"like-in-a-book\" f orm, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "matrix(U) * ma trix(CM) = matrix(`U CM`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'m atrixG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*F*-F&6#7%7#&%#cmG6#\"#67#&F36 #\"#@7#&F36#\"#JF*F." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 261 "" 0 "" {TEXT 307 5 "* * *" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 4 "N.B." }{TEXT -1 20 " The product of a " }{TEXT 261 10 "row matrix" }{TEXT -1 11 " and the " }{TEXT 262 11 "unit matrix" }{TEXT -1 10 " is the " } {TEXT 301 6 "matrix" }{TEXT -1 2 " " }{TEXT 302 9 "unchanged" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 347 1 "(" } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 348 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 349 1 ")" }{TEXT -1 2 " " }{TEXT 291 10 "row matrix " }{TEXT -1 3 " [" }{TEXT 292 2 "RM" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "RM := matrix(1, 3, [rm[11], \+ rm[12], rm[13]]) : RM = matrix(RM) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#RMG-%'matrixG6#7#7%&%#rmG6#\"#6&F+6#\"#7&F+6#\"#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "an d a " }{TEXT 350 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 351 3 " \+ \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 352 1 ")" }{TEXT -1 2 " " }{TEXT 299 11 "unit matrix" }{TEXT -1 3 " [" }{TEXT 298 1 "U" }{TEXT -1 2 "]." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 14 "The product [" }{TEXT 294 2 "RM" }{TEXT -1 3 "] [ " }{TEXT 293 1 "U" }{TEXT -1 21 "] is the following " }{TEXT 353 1 " (" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 354 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 355 1 ")" }{TEXT -1 13 " row matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`RM U` := multiply(RM, U) : RM*U \+ = matrix(`RM U`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#RMG\"\"\"% \"UGF&-%'matrixG6#7#7%&%#rmG6#\"#6&F.6#\"#7&F.6#\"#8" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This op eration may be displayed in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "matrix(RM) * matrix(U) = matrix(`RM U`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%'matrixG6#7#7%&%#rmG6# \"#6&F+6#\"#7&F+6#\"#8\"\"\"-F&6#7%7%F4\"\"!F97%F9F4F97%F9F9F4F4F%" }} }{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 308 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 365 4 "N.B." }{TEXT -1 39 " The determinant of \+ a unit matrix is " }{XPPMATH 20 "6#%&unityG" }{TEXT -1 1 "." }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 366 11 "determinant" }{TEXT -1 36 " of a m atrix, refer to Unit (11). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 275 "" 0 "" {TEXT 364 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 309 4 "N.B." } {TEXT -1 19 " A unit matrix is " }{TEXT 316 3 "not" }{TEXT -1 13 " ch anged by " }{TEXT 310 13 "transposition" }{TEXT -1 6 " or " }{TEXT 311 9 "inversion" }{TEXT -1 1 "." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "[ Refer to Unit (10) for t he " }{TEXT 367 9 "transpose" }{TEXT -1 40 " of a matrix and to Unit (14) for the " }{TEXT 368 7 "inverse" }{TEXT -1 16 " of a matrix. ] " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 314 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 4 "N.B." }{TEXT -1 23 " A unit matrix is an " }{TEXT 313 10 "orthogonal" }{TEXT -1 9 " matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "[ For the " }{TEXT 369 10 "orthogonal" }{TEXT -1 31 " matrix, refer t o Unit (17). ]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 363 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (10) fo r \"" }{TEXT 362 25 "The transpose of a matrix" }{TEXT -1 2 "\"." }}} {EXCHG {PARA 264 "" 0 "" {TEXT 361 67 "------------------------------- ------------------------------------" }}}}{MARK "79 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }