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0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 274 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 }} {SECT 0 {EXCHG {PARA 265 "" 0 "" {TEXT 333 39 "MATRICES AND MATRIX OPE RATIONS: Unit 20" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 266 "" 0 "" {TEXT 335 23 "Dr. Wlodzislaw Kostecki" }}{PARA 267 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 268 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 269 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 270 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" {TEXT 334 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 272 "" 0 "" {TEXT 336 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT 337 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(20)" }{TEXT 259 1 " " }{TEXT 258 17 "Trace of a matrix " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 341 10 "OBJECTIVES" }{TEXT 342 1 ":" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 343 1 "\225" }{TEXT -1 17 " To define the " }{TEXT 345 5 "trace" }{TEXT -1 21 " of a squar e matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 344 1 "\225" }{TEXT -1 59 " To specify and illustrate properties of the matrix trace." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "restart : interface(warnlevel=0) : with(linalg, i nverse, trace) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The trace of a " }{TEXT 319 6 "square" } {TEXT -1 76 " matrix is the sum of the elements of the principal diag onal of the matrix." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Consider a " }{TEXT 320 1 "(" } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 260 3 " \327 " }{XPPEDIT 18 0 "3" " 6#\"\"$" }{TEXT 321 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 261 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 26 "The trace of the \+ matrix, " }{TEXT 262 5 "Trace" }{TEXT -1 1 "[" }{TEXT 263 1 "A" } {TEXT -1 6 "], is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`trac e(A)` := trace(A) : Trace(A) = `trace(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#%\"AG,(&%\"aG6#\"#6\"\"\"&F*6#\"#AF-&F*6#\" #LF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 26 "For instance, consider a " }{TEXT 322 1 "(" }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 264 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 323 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 265 1 "A" }{TEXT -1 43 "] containing numerical elements as follows:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "A := matrix(3, 3, [4, 2, 1, 1, -1, 0, 1, 1, 3] ) : A = matrix(A) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG-%'matr ixG6#7%7%\"\"%\"\"#\"\"\"7%F,!\"\"\"\"!7%F,F,\"\"$" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The trace of [" }{TEXT 266 1 "A" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "`trace(A)` := trace(A) : Trace(A) = `trace(A)` ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#%\"AG\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 267 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 4 "N.B." }{TEXT -1 27 " If two square matrices, [" } {TEXT 269 1 "A" }{TEXT -1 7 "] and [" }{TEXT 270 1 "B" }{TEXT -1 89 "] , are of the same order, then the trace of their sum is equal to the s um of their traces" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 271 6 "Trace(" } {TEXT -1 1 "[" }{TEXT 272 1 "A" }{TEXT -1 1 "]" }{TEXT 324 3 " + " } {TEXT -1 1 "[" }{TEXT 273 1 "B" }{TEXT -1 1 "]" }{TEXT 325 9 ") = Trac e" }{TEXT -1 1 "[" }{TEXT 274 1 "A" }{TEXT -1 1 "]" }{TEXT 326 8 " + T race" }{TEXT -1 1 "[" }{TEXT 275 1 "B" }{TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "F or example, consider the matrix [" }{TEXT 276 1 "A" }{TEXT -1 23 "] as above and matrix [" }{TEXT 277 1 "B" }{TEXT -1 10 "] given as" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "B := matrix(3, 3, [8, 2, 1, \+ 1, 3, 0, 4, 2, 3]) : B = matrix(B) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"BG-%'matrixG6#7%7%\"\")\"\"#\"\"\"7%F,\"\"$\"\"!7%\"\"%F+F. " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "(a) The trace of the sum matrix, " }{TEXT 278 6 "Trace( " }{TEXT -1 1 "[" }{TEXT 279 1 "A" }{TEXT -1 1 "]" }{TEXT 327 3 " + " }{TEXT -1 1 "[" }{TEXT 280 1 "B" }{TEXT -1 1 "]" }{TEXT 328 1 ")" } {TEXT -1 5 ", is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "`trace (A+B)` := trace(A + B) : Trace(A + B) = `trace(A+B)` ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&TraceG6#,&%\"AG\"\"\"%\"BGF)\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "( b) The sum of traces of both matrices, " }{TEXT 281 5 "Trace" }{TEXT -1 1 "[" }{TEXT 282 1 "A" }{TEXT -1 1 "]" }{TEXT 329 8 " + Trace" } {TEXT -1 1 "[" }{TEXT 283 1 "B" }{TEXT -1 6 "], is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "`trace(A)+trace(B)` := trace(A) + trace(B ) : Trace(A) + Trace(B) = `trace(A)+trace(B)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%&TraceG6#%\"AG\"\"\"-F&6#%\"BGF)\"#?" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 284 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 4 "N.B." }{TEXT -1 27 " If two square matrices, [" } {TEXT 286 1 "A" }{TEXT -1 7 "] and [" }{TEXT 287 1 "B" }{TEXT -1 111 " ], are of the same order, then the trace of their product does not dep end on the order in matrix multiplication" }}}{EXCHG {PARA 262 "" 0 " " {TEXT 288 6 "Trace(" }{TEXT -1 1 "[" }{TEXT 289 1 "A" }{TEXT -1 3 "] [" }{TEXT 290 1 "B" }{TEXT -1 1 "]" }{TEXT 330 10 ") = Trace(" } {TEXT -1 1 "[" }{TEXT 291 1 "B" }{TEXT -1 3 "] [" }{TEXT 292 1 "A" } {TEXT -1 1 "]" }{TEXT 331 1 ")" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Exemplarily, consider the sam e matrices [" }{TEXT 293 1 "A" }{TEXT -1 7 "] and [" }{TEXT 294 1 "B" }{TEXT -1 23 "] that were used above." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "(a) The trace of \+ the product [" }{TEXT 295 1 "A" }{TEXT -1 3 "] [" }{TEXT 296 1 "B" } {TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "`trace (AB)` := trace(evalm(A &* B)) : Trace(`A B`) = `trace(AB)` ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#%$A~BG\"#Z" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "( b) The trace of the product [" }{TEXT 297 1 "B" }{TEXT -1 3 "] [" } {TEXT 298 1 "A" }{TEXT -1 5 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "`trace(BA)` := trace(evalm(B &* A)) : Trace(`B A`) \+ = `trace(BA)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#%$B~AG \"#Z" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 299 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 300 4 "N.B." }{TEXT -1 13 " If matrix [ " }{TEXT 302 1 "B" }{TEXT -1 55 "] is an invertible matrix of the same order as matrix [" }{TEXT 301 1 "A" }{TEXT -1 59 "], then the trace o f the product matrix of the inverse of [" }{TEXT 309 1 "B" }{TEXT -1 7 "] and [" }{TEXT 310 1 "A" }{TEXT -1 3 "] [" }{TEXT 311 1 "B" } {TEXT -1 28 "] is equal to the trace of [" }{TEXT 312 1 "A" }{TEXT -1 1 "]" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 304 10 "Trace\{(Inv" }{TEXT -1 1 "[" }{TEXT 305 1 "B" }{TEXT -1 1 "]" }{TEXT 307 1 ")" }{TEXT -1 2 " [" }{TEXT 306 1 "A" }{TEXT -1 3 "] [" }{TEXT 308 1 "B" }{TEXT -1 1 "]" }{TEXT 332 9 "\} = Trace" }{TEXT -1 1 "[" }{TEXT 303 1 "A" } {TEXT -1 1 "]" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Exemplarily, consider the same matrices [ " }{TEXT 313 1 "A" }{TEXT -1 7 "] and [" }{TEXT 314 1 "B" }{TEXT -1 12 "] as before." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(a) The trace of the product matrix of th e inverse of [" }{TEXT 315 1 "B" }{TEXT -1 7 "] and [" }{TEXT 316 1 "A " }{TEXT -1 3 "] [" }{TEXT 317 1 "B" }{TEXT -1 4 "] is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`trace((inv(B) AB)` := trace(evalm( inverse(B) &* A &* B)) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Trace(Inv(B)*A*B) = `trace((inv(B) AB)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6#*(-%$InvG6#%\"BG\"\"\"%\"AGF,F+F,\"\"'" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "(b) The trace of [" }{TEXT 318 1 "A" }{TEXT -1 4 "] is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "`trace(A)` := trace(A) : T race(A) = `trace(A)` ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&TraceG6# %\"AG\"\"'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 274 "" 0 "" {TEXT 340 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (21) for \"" }{TEXT 339 40 "Eigenvalues and eigenvectors of matrices" }{TEXT -1 2 " \"." }}}{EXCHG {PARA 264 "" 0 "" {TEXT 338 67 "----------------------- --------------------------------------------" }}}}{MARK "41 0 0" 11 } {VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }