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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "MATRIX OPERATIONS" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We begin \+
by loading the linalg package which defines most of the matrix functio
ns used here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linal
g):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 333 "In Maple, a matrix can be
 defined by feeding a list of lists to the function \"matrix\".  An mx
n matrix (with  m  rows and  n  columns)  corresponds to a list of m e
lements (a column vector) each of which is a list of  n  numbers (the \+
rows of the matrix).  For example, we specify the 2x3 matrix  M  with \+
rows  [1,1,1] and  [2,1,3]  by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "M := matrix([[1,1,1],[2,1,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"MG-%'matrixG6#7$7%\"\"\"F*F*7%\"\"#F*\"\"$" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 132 "Alternatively, we can specify the number of ro
ws and columns of the matrix, and then provide the elements in order i
n a single list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "M := ma
trix(2,3,[1,1,1,2,1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'m
atrixG6#7$7%\"\"\"F*F*7%\"\"#F*\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 123 "Individual elements of a matrix can be addressed using two ind
ices, the first indicating the row and the second the column:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "M[2,1];  M[2,3];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "The transpose T of the ma
trix  M  is the  3x2   matrix whose rows are the columns of  M.  It is
 calculated using the \"transpose\" function: " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "T := transpose(M);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"TG-%'matrixG6#7%7$\"\"\"\"\"#7$F*F*7$F*\"\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 450 "The product, AB, of two matrices \+
is calculated using the binary operator \"&*\"; that is we calculate  \+
A &* B.  Of course, the number of columns of A must be equal to the nu
mber of rows of B.  The resulting matrix product will be left in symbo
lic form (i.e. as A &* B) unless we force its evaluation by using the \+
function \"evalm\".  Here we calculate the product of  M  and its tran
spose T, and then evaluate it with evalm.  The result is a  2x2  matri
x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P := M &* T;  evalm(P)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%#&*G6$%\"MG%\"TG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"$\"\"'7$F)\"#9" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "On the other hand,  T &8* M  is a
 3x3 matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Q := evalm(
T &* M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'matrixG6#7%7%\"\"
&\"\"$\"\"(7%F+\"\"#\"\"%7%F,F/\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 146 "Observe that both  M &* T  and  T &* M  are symmetric, square \+
matrices.  (This is always true of the matrix product of a matrix and \+
its transpose." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 112 "The determinant of  a square matrix (one with equal numb
ers of rows and columns) is given by the \"det\" function:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A := matrix([[1,1,1],[2,1,3],[5,-1,
-2]]);  det(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7
%7%\"\"\"F*F*7%\"\"#F*\"\"$7%\"\"&!\"\"!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The invers
e of a nonsingular square matrix  A  can be calculated as \"inverse(A)
\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Ainv := inverse(A);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AinvG-%'matrixG6#7%7%#\"\"\"\"#
8F*#\"\"#F,7%#\"#>F,#!\"(F,#!\"\"F,7%F2#\"\"'F,F4" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "evalm(A &* Ainv); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "When a matrix is multiplied on the
 right by a list, Maple treats the list as a column vector." }}{PARA 
0 "" 0 "" {TEXT -1 89 "When a matrix is multiplied on the left by a li
st, Maple treats the list as a row vector." }}{PARA 0 "" 0 "" {TEXT 
-1 56 "In either case, the result can be simplified by  \"evalm\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalm(A &* [x,y,z]);  evalm(
[x,y,z] &* A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,(%\"x
G\"\"\"%\"yGF)%\"zGF),(F(\"\"#F*F)F+\"\"$,(F(\"\"&F*!\"\"F+!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,(%\"xG\"\"\"%\"yG\"\"#%
\"zG\"\"&,(F(F)F*F)F,!\"\",(F(F)F*\"\"$F,!\"#" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 179 "A set of  n  linear equations in  n  variables can be
 written in the form  AX=B where A is an nxn matrix and X and B are co
lumn n-vectors.  Thus the solution can be calculated as " }}{PARA 0 "
" 0 "" {TEXT -1 40 "X := Ainv &* B.  For example, the system" }}{PARA 
0 "" 0 "" {TEXT -1 30 "        x  +  y  +    z   =  2" }}{PARA 0 "" 0 
"" {TEXT -1 27 "      2x  +  y  +  3z  =  9" }}{PARA 0 "" 0 "" {TEXT 
-1 26 "      5x  -  y  - 2z  =  1" }}{PARA 0 "" 0 "" {TEXT -1 85 "has \+
has the matrix A  defined above, and  B = [2,9,1].  The solution of th
e system is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "X := evalm(A
inv &* [2,9,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'vectorG6#
7%\"\"\"!\"#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "that is,  x \+
= 1,  y = -2,  z = 3.  Maple provides a simpler way of solving the sys
tem  AX = B; " }}{PARA 0 "" 0 "" {TEXT -1 46 "we just need to use the \+
function linsolve(A,B)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "X
 := linsolve(A,[2,9,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'v
ectorG6#7%\"\"\"!\"#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "\"l
insolve\" is better at solving linear systems than is matrix inversion
, since it can solve some systems for which the matrix is singular:  C
onsider the systems" }}{PARA 0 "" 0 "" {TEXT -1 85 "    x   +     y  =
  1                                                 x   +   y  =  1" }
}{PARA 0 "" 0 "" {TEXT -1 80 "  2x   +  2y  =  2                      \+
                          2x  + 2y  =  1" }}{PARA 0 "" 0 "" {TEXT -1 
97 "The first has a one parameter family of solutions  x = t,  y = 1-t
.  The second has no solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 54 "L := matrix([[1,1],[2,2]]); C1 := [1,2];  C2 := [1,1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'matrixG6#7$7$\"\"\"F*7$\"\"#F
," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G7$\"\"\"\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#C2G7$\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "X := linsolve(L,C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"XG-%'vectorG6#7$,&\"\"\"F*&%#_tG6#F*!\"\"F+" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "X := linsolve(L,C2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"XG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "Now \+
X is undefined, indicating no solutions for LX=C2.  But the matrix \"i
nverse(L)\" does not exist, so the solution of LX=C1 cannot be found a
s X := inverse(L) &* C1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "
X := inverse(L) &* C1;" }}{PARA 8 "" 1 "" {TEXT -1 35 "Error, (in inve
rse) singular matrix" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "The lina
lg package has procedures for finding the eigenvalues and eigenvectors
 of matrices.  For a real symmetric matrix, the eigenvalues are always
 real.    " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "K := matrix([
[3,1,-1],[1,4,1],[-1,1,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG
-%'matrixG6#7%7%\"\"$\"\"\"!\"\"7%F+\"\"%F+7%F,F+F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "eigenvals(K); evalf(%);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%\"\"%,&\"\"$\"\"\"*$-%%sqrtG6#F%\"\"\"F&,&F%F&F'!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"%\"\"!$\"+330KZ!\"*$\"+#>\\
zE\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(K);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%,&\"\"$\"\"\"*$-%%sqrtG6#F%\"\"
\"F&F&<#-%'vectorG6#7%F&,&F&F&F'F&F&7%,&F%F&F'!\"\"F&<#-F.6#7%F&,&F&F&
F'F4F&7%\"\"%F&<#-F.6#7%F4\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "35 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }