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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 30 "\011\011\011\011\011Some
 Basic Linear Algebra" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 304 26 "Versio
n .8 for Maple V R7." }}}{EXCHG {PARA 256 "" 0 "" {TEXT 257 107 "Load \+
the linear algebra package. Below is the list of routines. You can get
\nhelp on any of them by typing ?" }{TEXT 258 14 "command_name  " }
{TEXT 259 33 "and then hitting the <ENTER> key." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning
, the protected names norm and trace have been redefined and unprotect
ed\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmi
dtG%,JordanBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addrowG%
$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG%&basisG%'bezoutG%,b
lockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%)colspaceG%
(colspanG%*companionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG%)d
efiniteG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eigenva
lsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG%,e
xponentialG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*frobenius
G%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(her
miteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*innerpr
odG%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'jor
danG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&mi
norG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nullsp
aceG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+randvector
G%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarm
ulG%-singularvalsG%&smithG%,stackmatrixG%*submatrixG%*subvectorG%)sumb
asisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transposeG%,
vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 29 "Entering Matrices and Vectors" }}{EXCHG 
{PARA 256 "" 0 "" {TEXT 260 22 "Enter a 2 by 3 matrix:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "A1 := matrix(2,3,[1,2,3,4,5,6]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7$7%\"\"\"\"\"#\"\"$7%\"\"%\"
\"&\"\"'" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 261 78 " These are viewed \+
as column vectors even though they are written horizontally." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "v1 := vector([5,3,1]); v2 := vector([3,4]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G-%'vectorG6#7%\"\"&\"\"$\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G-%'vectorG6#7$\"\"$\"\"%
" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Matrix Multiplication, Eval
m, Transpose, Determinant, Inverses" }}{EXCHG {PARA 256 "" 0 "" {TEXT 
262 101 "Enclose matrix computations within the function call evalm.\n
&* is the matrix multiplication operator." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "evalm(A1 &*v1 + v2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'vectorG6#7$\"#<\"#X" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "A2 := evalm(A1 &* transpose(A1) );" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#A2G-%'matrixG6#7$7$\"#9\"#K7$F+\"#x" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT 263 53 "Ordinary scalar multiplication is indicated with a \+
*." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(2*A2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"#G\"#k7$F)\"$a\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(A2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#a" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 264 65 "Calcula
te the inverse of a matrix. \nA2^(-1) would also work here." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "A2_inv := inverse(A2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'A2_invG-%'matrixG6#7$7$#\"#x\"#a#!#;\"#F7$F-#\"\"(
F/" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "The Map Function - Applyi
ng a function to every entry of a matrix." }}{EXCHG {PARA 256 "" 0 "" 
{TEXT 265 26 "Convert to floating point." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "map(evalf, A2_inv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'matrixG6#7$7$$\"+Ef#fU\"!\"*$!+Ef#f#f!#57$F+$\"+$f#f#f#F-" }}}}
{EXCHG {PARA 259 "" 0 "" {TEXT 266 84 "See the worksheet \"Entering Ma
trices\" for more information on constructing matrices." }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 48 "Bases, Rank and Subspaces Associated to a
 Matrix" }}{EXCHG {PARA 256 "" 0 "" {TEXT 305 37 "\nCompute a basis fo
r the range of A1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "colspace(tran
spose(A1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7%\"\"\"
\"\"!!\"\"-F%6#7%F)F(\"\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 267 1 " \+
" }}{PARA 256 "" 0 "" {TEXT 268 21 "Basis for the kernel." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "nullspace(matrix(3,3,[[1,2,3],[2,4,6],[3,6,9
]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7%!\"#\"\"\"\"
\"!-F%6#7%!\"$F*F)" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 269 1 " " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rank(A2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 270 58 "Find a
 basis for the subspace spanned by a set of vectors." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "basis(\{vector([1,0,1]),vector([1,0,5]),vector([
3,0,1])\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7%\"\"$\"
\"!\"\"\"-F%6#7%F*F)F*" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 308 25 "Redu
ced row echelon form." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rref(A1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7%\"\"\"\"\"!!\"\"7%F)
F(\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Solution of a Linear
 System" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "linsolve(matrix(2
,2,[a,b,c,d]),vector([e,f]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ve
ctorG6#7$*&,&*&%\"bG\"\"\"%\"fGF+F+*&%\"eGF+%\"dGF+!\"\"F+,&*&%\"aGF+F
/F+F0*&%\"cGF+F*F+F+F0*&,&*&F3F+F,F+F0*&F5F+F.F+F+F+F1F0" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 29 "Dot Product and Cross Product" }}{EXCHG 
{PARA 256 "" 0 "" {TEXT 306 12 "Dot product." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "dotprod(v1, vector([1,4,8]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#D" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 271 21 "Cross p
roduct in R^3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "crossprod(v1,vect
or([1,0,0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!\"
\"\"!\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Rows, Columns, and \+
Submatrices" }}{EXCHG {PARA 256 "" 0 "" {TEXT 272 109 "Copy the all th
e entries of A1 into the matrix B. The upper left corner of A1\ngoes i
nto the (2,1) entry of B." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B:= ma
trix(3,3);copyinto(A1,B,2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"B
G-%&arrayG6%;\"\"\"\"\"$F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ma
trixG6#7%7%&%\"?G6$\"\"\"F+&F)6$F+\"\"#&F)6$F+\"\"$7%F+F.F17%\"\"%\"\"
&\"\"'" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 273 12 "Row 1 of A1." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "row(A1,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT 274 15 "Column 1 of A1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "col(A1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$\"\"\"
\"\"%" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 275 56 "Construct a submatrix
 of A using rows 1..2 and column 1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "submatrix(A1,1..2,1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat
rixG6#7$7#\"\"\"7#\"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 5 "Norms
" }}{EXCHG {PARA 256 "" 0 "" {TEXT 276 105 "The norm function uses abs
olute value signs which is somewhat inconvenient for algebraic \nsimpl
ification." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "norm(vector([x,y]),2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$)-%$absG6#%\"xG\"
\"#\"\"\"F/*$)-F+6#%\"yGF.F/F/F/" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
277 61 "So it may be better top define your own vector norm function:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "vnorm := u -> sqrt(dotprod(u,u,
'orthogonal'));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&vnormGf*6#%\"uG6
\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%(dotprodG6%9$F2.%+orthogonalGF(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "vnorm(vector([x,y]),2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"
\"F,*$)%\"yGF+F,F,F," }}}{EXCHG {PARA 256 "" 0 "" {TEXT 278 37 "The de
fault norm is the infinity one." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
norm(vector([x,y]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$maxG6$-%$ab
sG6#%\"yG-F'6#%\"xG" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 87 "Using Ma
ple's Solve Command to find the set of matrices which\ncommute with a \+
given one." }}{EXCHG {PARA 256 "" 0 "" {TEXT 279 198 "Matrix equations
 can be converted into sets of equations and then solved. \nFor exampl
e consider the problem of finding all matrices B which commute with th
e matrix A2 above (i.e. A2 B - B A2 = 0) :\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "B := matrix(2,2,[a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mat_with_unknowns := evalm( \+
A2 &* B - B &* A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2mat_with_unk
nownsG-%'matrixG6#7$7$,&%\"cG\"#K*&F,\"\"\"%\"bGF.!\"\",(F/!#j*&F,F.%
\"dGF.F.*&F,F.%\"aGF.F07$,(F6F,*&\"#jF.F+F.F.*&F,F.F4F.F0,&F/F,*&F,F.F
+F.F0" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 280 316 " Now we solve for th
e unknown entries a,b,c, and d.\n(This relies on the fact that Maple's
 solve command ordinarily wants a set of equations in its\nfirst argum
ent. But if given a first argument which is a set of algebraic equatio
ns,\nit interprets these as a set of equations by viewing each express
ion as equal to 0.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "soln1 := sol
ve(convert(mat_with_unknowns,set),\{a,b,c,d\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&soln1G<&/%\"dGF'/%\"aGF)/%\"bG,&F'#\"#K\"#j*&#F.F/\"
\"\"F)F2!\"\"/%\"cGF," }}}{EXCHG {PARA 256 "" 0 "" {TEXT 281 68 "This \+
globally assigns b and a to have the values mentioned in soln1." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assign(soln1);" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT 282 167 " Even though print(B) does not show the sub
stitutes expressions, the entries of B\n (e.g. the upper left corner a
) really does have the value specified in the solution." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "B[1,1]; print(B);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"aG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$
7$%\"aG%\"bG7$%\"cG%\"dG" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 283 37 " F
ully evaluate all the entries of B." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "print(map(eval,B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6
#7$7$%\"aG,&%\"dG#\"#K\"#j*&#F,F-\"\"\"F(F0!\"\"7$F)F*" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 32 "Gram Schmidt, Matrix Exponential" }}
{EXCHG {PARA 256 "" 0 "" {TEXT 284 129 "Convert a list of vectors into
 a list of orthogonal vectors. The first k vectors in\neach list (for \+
any k) span the same subspace." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "G
ramSchmidt([vector([1,1,1]),vector([1,2,3]),vector([1,4,5])]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7%\"\"\"F%F%7%!\"\"\"\"!F%7%#F'\"\"$
#\"\"#F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 307 18 "matrix exponential
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "exponential(matrix(2,2,[0,theta
,-theta,0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$-%$co
sG6#%&thetaG-%$sinGF*7$,$F,!\"\"F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 43 "Calculus of R^n and Vector Valued Functions" }}{EXCHG {PARA 
256 "" 0 "" {TEXT 285 50 "The derivative of a function of several vari
ables." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "deriv := jacobian([exp(x)
*cos(y),exp(x)*sin(y)],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&d
erivG-%'matrixG6#7$7$*&-%$expG6#%\"xG\"\"\"-%$cosG6#%\"yGF/,$*&F+F/-%$
sinGF2F/!\"\"7$F5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eval
_at_1_2 := a -> subs(x=1,y=2,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
,eval_at_1_2Gf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%%subsG6%/%\"xG\"\"\"
/%\"yG\"\"#9$F(F(F(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 286 48 " A func
tion to evaluate anything at (x,y)=(1,2)." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "map(eval_at_1_2,deriv);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7$7$*&-%$expG6#\"\"\"F,-%$cosG6#\"\"#F,,$*&F)F,-%$si
nGF/F,!\"\"7$F2F(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 287 51 " Substitu
te (x,y) =  (1,2) for each entry of deriv." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "r := (x^2+y^2+z^2)^(1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG*$-%%sqrtG6#,(*$)%\"xG\"\"#\"\"\"F.*$)%\"yGF-F.F.
*$)%\"zGF-F.F.F." }}}{EXCHG {PARA 256 "" 0 "" {TEXT 288 23 "Curl of a \+
vector field." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "curl([x/r^3,y/r^3,
z/r^3],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"
!F'F'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Eigenvalues, Eigenvect
ors, and the SVD" }}{EXCHG {PARA 256 "" 0 "" {TEXT 289 10 "A list of \+
" }{TEXT 290 8 "symbolic" }{TEXT 291 14 "  eigenvalues." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "evals_of_A2 := eigenvals(A2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%,evals_of_A2G6$,&#\"#\"*\"\"#\"\"\"*&#F*F)F*-%%s
qrtG6#\"%l!)F*F*,&F'F**&#F*F)F**$F-F*F*!\"\"" }}}{EXCHG {PARA 256 "" 
0 "" {TEXT 292 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evals_of_A2[
1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"#\"*\"\"#\"\"\"*&#F'F&F'-%
%sqrtG6#\"%l!)F'F'" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 293 43 "Extract \+
the first eigenvalue from the list." }}{PARA 256 "" 0 "" {TEXT 294 56 
"Convert the first eigenvalue to a floating point number." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "evalf(evals_of_A2[1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+_sES!*!\")" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 295 
71 "Convert evals_of_A2 to a list, and change each entry to floating p
oint." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "map(evalf, [evals_of_A2]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+_sES!*!\")$\")[FtfF&" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 296 107 "Just doing \n\011\011 map(eval
f, evals_of_A2);\nwould result in an error.\nDirectly calculate numeri
cal eigenvalues." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(Eigenvals
(A2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+:ZFtf!#5$
\"+`sES!*!\")" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 297 1 " " }}{PARA 
256 "" 0 "" {TEXT 298 177 "Get the eigenvectors as well as the eigenva
lues. Assign the eigenvectors of A2 to the \ncolumns of w.\n(NOTE: Use
 a fresh variable instead of w if you run into difficulties here.)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(Eigenvals(A2,w));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\")[Ftf!\")$\"+asES!*F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(w);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7$7$$\"+,ylB#*!#5$\"+Iq<jQF*7$$!+Iq<jQF*F
(" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 299 1 " " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "evalm(inverse(w) &* A2 &* w); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$$\"+'puK(f!#5$\"#lF*7$$\"\"!F/$\"+`sES!
*!\")" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 300 39 "Check the eigenvector
s and eigenvalues." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "charpoly(eval
m(transpose(A1)&*A1),lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)
%'lambdaG\"\"$\"\"\"F(*&\"#\"*F()F&\"\"#F(!\"\"*&\"#aF(F&F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singularvals(A1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$,$*$-%%sqrtG6#,&\"$#=\"\"\"*&\"\"#F+-F'6#
\"%l!)F+F+F+#F+F-,$*$-F'6#,&F*F+*&F-F+F.F+!\"\"F+F1" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT 301 131 "Compute the singular value decomposition of
 A1. (So transpose(U) A V will have nearly 0 \nentries except for thes
e on the diagonal.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(Svd(A1
,U,V));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+,?.3&*!\"
*$\"+djpGx!#5" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 302 39 "The orthogona
l matrices computed above." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print
(U,V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7$7$$!+Jq<jQ!#5$
!+,ylB#*F*7$F+$\"+Jq<jQF*-F$6#7%7%$!+N8n'G%F*$\"+'3R'f!)F*$\"+0H[#3%F*
7%$!+)=pIm&F*$\"+TT#Q7\"F*$!+4e'\\;)F*7%$!+TqYRqF*$!+/3*>\"eF*F8" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 303 21 " Check the SVD above." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(transpose(U) &* A1 &* V);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7%$\"++?.3&*!\"*$\"\"\"F*$\"\"
!F.7%$!\"&!#5$\"+]jpGxF2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "15 14 0
 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }