Answers, using Maple (you should do
Answers, using Maple (you should do the problems by hand, but I wanted to show
you how to check your answers using Maple)
with(linalg):
A := matrix([[1,1],[0,0]]);
B := matrix([[5,6],[-1,-2]]);
C := matrix([[2,-8],[1,-4]]);
d := matrix([[2,2,1],[0,1,2],[0,0,-1]]);
E := matrix([[0,1,2],[0,0,1],[0,0,0]]);
F := matrix([[1,1],[0,0]]);
G := matrix([[0,-1],[0,0]]);
H := matrix([[1,0],[0,0]]);
Warning, new definition for norm
Warning, new definition for trace
eigenvectors(A);
eigenvectors(B);
eigenvectors(C);
eigenvectors(d);
[1, 1, {[1, 0]}], [0, 1, {[-1, 1]}] |
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[-1, 1, {[-1, 1]}], [4, 1, {[-6, 1]}] |
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[-2, 1, {[2, 1]}], [0, 1, {[4, 1]}] |
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[1, 1, {[-2, 1, 0]}], [-1, 1, {[1, -3, 3]}], [2, 1, {[1, 0, 0]}] |
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S:=transpose([[-1,1],[1,0]]);
L:=matrix([[0,0],[0,1]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[1,-1],[-6,1]]);
L:=matrix([[-1,0],[0,4]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[2,1],[4,1]]);
L:=matrix([[-2,0],[0,0]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[-2,1,0],[1,-3,3],[1,0,0]]);
L:=matrix([[1,0,0],[0,-1,0],[0,0,2]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[-1,1],[1,0]]);
L:=matrix([[0,0],[0,1]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[1,-1],[-6,1]]);
L:=matrix([[(-1)^6,0],[0,4^6]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[2,1],[4,1]]);
L:=matrix([[(-2)^6,0],[0,0]]);
multiply(S,multiply(L,inverse(S)));
S:=transpose([[-2,1,0],[1,-3,3],[1,0,0]]);
L:=matrix([[1,0,0],[0,(-1)^6,0],[0,0,2^6]]);
multiply(S,multiply(L,inverse(S)));
I am cheating from now on. You should really use the diagonalizations which
you obtained above. I am using the exponential function in Maple just to get
the answers.
exponential(A,t);
exponential(B,t);
exponential(C,t);
exponential(d,t);
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é ê
ë
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- [1/ 5] e( - t) + [6/ 5] e(4 t) |
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[6/ 5] e(4 t) - [6/ 5] e( - t) |
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- [1/ 5] e(4 t) + [1/ 5] e( - t) |
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[6/ 5] e( - t) - [1/ 5] e(4 t) |
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ù ú
û
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é ê
ë
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[1/ 2] - [1/ 2] e( - 2 t) |
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ù ú
û
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é ê ê
ê ê ë
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[5/ 3] e(2 t) - 2 et + [1/ 3] e( - t) |
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ù ú ú
ú ú û
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(cheating again)
exponential(matrix([[0,1,2],[0,0,1],[0,0,0]]),t);
multiply(exponential(F,t),exponential(G,t));
exponential(H,t);
File translated from TEX by TTH, version 2.00.
On 24 Feb 1999, 16:47.