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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 54 "Linear Algebra, Infinite Dimens
ional Spaces, and Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 23 "Jim Herod, Georgia Tech" }}{PARA 0 "" 0 "" {TEXT -1
21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 10 "Section 1:" }
{TEXT -1 29 " A Decomposition for Matrices" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 " This section provides a p
roof for the Jordan Canonical Form for a square matrix. It establishes
that if A is a matrix then A can be written as" }}{PARA 0 "" 0 ""
{TEXT -1 45 " A = " }{XPPEDIT
18 0 "sum(lambda[i]*P[i]+N[i],i=1..k)" "6#-%$sumG6$,&*&&%'lambdaG6#%\"
iG\"\"\"&%\"PG6#F+F,F,&%\"NG6#F+F,/F+;\"\"\"%\"kG" }}{PARA 0 "" 0 ""
{TEXT -1 6 "where " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1
125 "'s are eigenvalues for A, the P's are projection matrices, and th
e N's are nilpotent matrices. A projection matrix satisfies " }
{XPPEDIT 18 0 "P^2=P" "6#/*$%\"PG\"\"#F%" }{TEXT -1 71 " and some powe
r of a nilpotent matrix N is zero. Here are two examples." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 80 "P:=matrix([[3,0,-2],[-1,1,1],[3,0,-2]]);\nN:=
matrix([[-1,1,1],[0,0,0],[-1,1,1]]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 23 "evalm(P^2);\nevalm(N^2);" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 " We find the Jordan Form fo
r the following matrix with the methods of the notes." }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 41 "A:=matrix([[-5,1,3],[1,-2,-1],[-4,1,2]]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C:=charpoly(A,lambda);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(C);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "convert(1/%,parfrac,lambda);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Term1:=op(1,%);\nTerm2:=simp
lify(op(2,%%)+op(3,%%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47
"Part1:=factor(Term1*C);\nPart2:=factor(Term2*C);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "is(1=Part1+Part2);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 65 "P1:=evalm(subs(lambda=A,Part1));\nP2:=evalm(subs
(lambda=A,Part2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Iden:
=diag(1,1,1);\nN1:=evalm(P1*(A+Iden));\nN2:=evalm(P2*(A+2*Iden));" }}}
{PARA 0 "" 0 "" {TEXT -1 59 "The following is a check to see that the \+
result is correct." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(
(-1*P1+N1) + (-2*P2+N2) );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "We could have \+
used Maple directly to find this form. The following command computes \+
what might better be called the diagonal form of A. It also computes a
matrix Q having the property that if J is the diagonal form then" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+
" }{XPPEDIT 18 0 "A = Q*J*Q^(
-1);" "6#/%\"AG*(%\"QG\"\"\"%\"JG\"\"\")F&,$\"\"\"!\"\"F'" }{TEXT -1
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "
Using this diagonal form and Q, we can get the Jordan form more direct
ly -- only, Maple did all the work." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "jordan(A,'Q');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 138 "p1:=evalm(Q&*diag(1,0,0)&*inverse(Q));\np2:=evalm(Q&*diag(0,1
,1)&*inverse(Q));\nN2:=evalm(Q&*matrix([[0,0,0],[0,0,1],[0,0,0]])&*inv
erse(Q));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 288 " Be aware tha
t this decomposition is not robust -- the job of factoring the charact
eristic polynomial is notorious! The decomposition sets the pattern fo
r many ideas to follow. We will seek to get decomposition of large cla
sses of linear transformation on infinite dimensional spaces." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 10 "Section \+
2:" }{TEXT -1 8 " Exp(tA)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 418 "In this section, we developed a method for comput
ing the exponential of a matrix using the Jordan Form. The importance \+
of the method for the context of this course is the structure of the r
epresentation. Using this representation, we will see later that class
es of partial differential equations can be put in the context of ordi
nary differential equations in the form Z ' = A Z, where Z has values \+
in a Hilbert Space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 62 "For the problems of this section, Maple can check the a
nswers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A:=matrix([[0,1],[1,0]]);\n
exptA:=exponential(A,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 54 }{VIEWOPTS 1 1 0
1 1 1803 }