# SECTION 9.5# Example 1with(linalg);

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A := matrix([[2,-3],[1,-2]]);

NiM+JSJBR0slJ21hdHJpeEc2IzckNyQiIiMhIiQ3JCIiIiEiI1EocHByaW50MDYi

I2 := matrix([[1,0],[0,1]]);# 2 by 2 identity matrix

NiM+JSNJMkdLJSdtYXRyaXhHNiM3JDckIiIiIiIhNyRGK0YqUShwcHJpbnQxNiI=

# Use I, as in text:interface(imaginaryunit=j); #Changes I from a reserved symbol to a valid variable name

NiNeIyIiIg==

I := evalm(I2);

NiM+JSJJR0slJ21hdHJpeEc2IzckNyQiIiIiIiE3JEYrRipRKHBwcmludDI2Ig==

A-r*I;

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evalm(%);

NiNLJSdtYXRyaXhHNiM3JDckLCYiIiMiIiIlInJHISIiISIkNyRGKiwmRilGLEYrRixRKHBwcmludDM2Ig==

det(%);

NiMsJiIiIiEiIiokKSUickciIiNGJEYk

solve(%,r);

NiQiIiIhIiI=

# Solve for eigenvectors:r1 := 1:evalm(A-r1*I);

NiNLJSdtYXRyaXhHNiM3JDckIiIiISIkRidRKHBwcmludDQ2Ig==

u := vector(2);

NiM+JSJ1Ry0lJmFycmF5RzYkOyIiIiIiIzci

(A-r1*I)&*u;

NiMtJSMmKkc2JCwmJSJBRyIiIiUiSUchIiIlInVH

evalm(%);

NiNLJSd2ZWN0b3JHNiM3JCwmJiUidUc2IyIiIkYrKiYiIiRGKyZGKTYjIiIjRishIiJGJ1EocHByaW50NTYi

eq := %[1] = 0;

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u[2] := s;

NiM+JiUidUc2IyIiIyUic0c=

u[1] := solve(eq,u[1]);

NiM+JiUidUc2IyIiIiwkKiYiIiRGJyUic0dGJ0Yn

u1 := evalm(u);

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r2 := -1;

NiM+JSNyMkchIiI=

u := vector(2);

NiM+JSJ1Ry0lJmFycmF5RzYkOyIiIiIiIzci

(A-r2*I)&*u;

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evalm(%);

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eq := %[1] = 0;

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u[2] := s;

NiM+JiUidUc2IyIiIyUic0c=

u[1] := solve(eq,u[1]);

NiM+JiUidUc2IyIiIiUic0c=

u2 := evalm(u);

NiM+JSN1MkdLJSd2ZWN0b3JHNiM3JCUic0dGKVEocHByaW50ODYi

# Example 2A := matrix([[1,2,-1],[1,0,1],[4,-4,5]]);

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I := matrix([[1,0,0],[0,1,0],[0,0,1]]);

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# Alternatively, we can obtain I as follows:?Idwith(Student[LinearAlgebra]);

Warning, the name GramSchmidt has been rebound 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

Id(3);

NiMtJSdSVEFCTEVHNiUiKl9fXk8iLSUnTUFUUklYRzYjNyU3JSIiIiIiIUYtNyVGLUYsRi03JUYtRi1GLCUnTWF0cml4Rw==

I := %;

NiM+JSJJRy0lJ1JUQUJMRUc2JSIqX19eTyItJSdNQVRSSVhHNiM3JTclIiIiIiIhRi83JUYvRi5GLzclRi9GL0YuJSdNYXRyaXhH

A-r*I;

NiMsJiUiQUciIiItJSdSVEFCTEVHNiUiKl9xZk4iLSUnTUFUUklYRzYjNyU3JSwkJSJyRyEiIiIiIUYyNyVGMkYvRjI3JUYyRjJGLyUnTWF0cml4R0Yl

evalm(%);

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det(%)=0;

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factor(%);

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solve(%%,r);

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r1 := 1;

NiM+JSNyMUciIiI=

u := vector(3);

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(A-r1*I)&*u;

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eq := evalm(%);

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eq1 := eq[1]=0; eq2 := eq[2]=0; eq3 := eq[3]=0;

NiM+JSRlcTFHLywmKiYiIiMiIiImJSJ1RzYjRihGKUYpJkYrNiMiIiQhIiIiIiE=NiM+JSRlcTJHLywoJiUidUc2IyIiIkYqJkYoNiMiIiMhIiImRig2IyIiJEYqIiIhNiM+JSRlcTNHLywoKiYiIiUiIiImJSJ1RzYjRilGKUYpKiZGKEYpJkYrNiMiIiNGKSEiIiomRihGKSZGKzYjIiIkRilGKSIiIQ==

# Interchange equations 1 and 2:temp := eq2; eq2 := eq1; eq1 := temp;

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eq1;eq2;eq3;

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eq3 := eq3 - 4*eq1;

NiM+JSRlcTNHLyIiIUYm

eq1;eq2;eq3;

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u[2] := s;

NiM+JiUidUc2IyIiIyUic0c=

u[3] := solve(eq2,u[3]);

NiM+JiUidUc2IyIiJCwkKiYiIiMiIiIlInNHRitGKw==

u[1] := solve(eq1,u[1]);

NiM+JiUidUc2IyIiIiwkJSJzRyEiIg==

u1 := evalm(u);

NiM+JSN1MUdLJSd2ZWN0b3JHNiM3JSwkJSJzRyEiIkYqLCQqJiIiIyIiIkYqRi9GL1EpcHByaW50MzE2Ig==

r2 := 2;

NiM+JSNyMkciIiM=

u := vector(3);

NiM+JSJ1Ry0lJmFycmF5RzYkOyIiIiIiJDci

(A-r2*I)&*u;

NiMtJSMmKkc2JCwmJSJBRyIiIiomIiIjRiglIklHRighIiIlInVH

eq := evalm(%);

NiM+JSNlcUdLJSd2ZWN0b3JHNiM3JSwoJiUidUc2IyIiIiEiIiomIiIjRi0mRis2I0YwRi1GLSZGKzYjIiIkRi4sKEYqRi0qJkYwRi1GMUYtRi5GM0YtLCgqJiIiJUYtRipGLUYtKiZGOkYtRjFGLUYuKiZGNUYtRjNGLUYtUSlwcHJpbnQzMzYi

eq1 := eq[1]=0; eq2 := eq[2]=0; eq3 := eq[3]=0;

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# Solving by using elementary row operations (Gaussian elimination):eq2 := eq2 + eq1;

NiM+JSRlcTJHLyIiIUYm

eq3 := eq3 +4*eq1;

NiM+JSRlcTNHLywmKiYiIiUiIiImJSJ1RzYjIiIjRilGKSZGKzYjIiIkISIiIiIh

eq1;eq2;eq3;

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u[2] := s;

NiM+JiUidUc2IyIiIyUic0c=

u[3] := solve(eq3,u[3]);

NiM+JiUidUc2IyIiJCwkKiYiIiUiIiIlInNHRitGKw==

u[1] := solve(eq1,u[1]);

NiM+JiUidUc2IyIiIiwkKiYiIiNGJyUic0dGJyEiIg==

u2 := evalm(u);

NiM+JSN1MkdLJSd2ZWN0b3JHNiM3JSwkKiYiIiMiIiIlInNHRiwhIiJGLSwkKiYiIiVGLEYtRixGLFEpcHByaW50MzQ2Ig==

r3 := 3;

NiM+JSNyM0ciIiQ=

u := vector(3);

NiM+JSJ1Ry0lJmFycmF5RzYkOyIiIiIiJDci

(A-r3*I)&*u;

NiMtJSMmKkc2JCwmJSJBRyIiIiomIiIkRiglIklHRighIiIlInVH

eq := evalm(%);

NiM+JSNlcUdLJSd2ZWN0b3JHNiM3JSwoKiYiIiMiIiImJSJ1RzYjRixGLCEiIiomRitGLCZGLjYjRitGLEYsJkYuNiMiIiRGMCwoRi1GLComRjZGLEYyRixGMEY0RiwsKComIiIlRixGLUYsRiwqJkY7RixGMkYsRjAqJkYrRixGNEYsRixRKXBwcmludDM1NiI=

eq1 := eq[1]=0; eq2 := eq[2]=0; eq3 := eq[3]=0;

NiM+JSRlcTFHLywoKiYiIiMiIiImJSJ1RzYjRilGKSEiIiomRihGKSZGKzYjRihGKUYpJkYrNiMiIiRGLSIiIQ==NiM+JSRlcTJHLywoJiUidUc2IyIiIkYqKiYiIiRGKiZGKDYjIiIjRiohIiImRig2I0YsRioiIiE=NiM+JSRlcTNHLywoKiYiIiUiIiImJSJ1RzYjRilGKUYpKiZGKEYpJkYrNiMiIiNGKSEiIiomRjBGKSZGKzYjIiIkRilGKSIiIQ==

# Solving as before:eq2 := eq2 +eq1/2;

NiM+JSRlcTJHLywmKiYiIiMiIiImJSJ1RzYjRihGKSEiIiomI0YpRihGKSZGKzYjIiIkRilGKSIiIQ==

eq3 := eq3 + 2*eq1;

NiM+JSRlcTNHLyIiIUYm

eq1 := eq1 + eq2;

NiM+JSRlcTFHLywmKiYjIiIiIiIjRikmJSJ1RzYjIiIkRikhIiIqJkYqRikmRiw2I0YpRilGLyIiIQ==

eq1;eq2;eq3;

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u[2] := s;

NiM+JiUidUc2IyIiIyUic0c=

u[3] := solve(eq2,u[3]);

NiM+JiUidUc2IyIiJCwkKiYiIiUiIiIlInNHRitGKw==

u[1] := solve(eq1,u[1]);

NiM+JiUidUc2IyIiIiwkJSJzRyEiIg==

u3 := evalm(u);

NiM+JSN1M0dLJSd2ZWN0b3JHNiM3JSwkJSJzRyEiIkYqLCQqJiIiJSIiIkYqRi9GL1EpcHByaW50MzY2Ig==

# Here's another way to find the eigenvalue-eigenvector pairs:eigenvects(A);

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# Example 3A := matrix([[2,-3],[1,-2]]);

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eigenvects(A);

NiQ3JSIiIkYkPCNLJSd2ZWN0b3JHNiM3JCIiJEYkUSlwcHJpbnQ0MTYiNyUhIiJGJDwjS0YnNiM3JEYkRiRRKXBwcmludDQyRiw=

x := c1*exp(1*t)*[3,1] + c2 *exp(-1*t)*[1,1];

NiM+JSJ4RywmKiglI2MxRyIiIi0lJGV4cEc2IyUidEdGKDckIiIkRihGKEYoKiglI2MyR0YoLUYqNiMsJEYsISIiRig3JEYoRihGKEYo

# Check that dx/dt = Ax:dxdt := map(diff,x,t);

NiM+JSVkeGR0RywuKiYtJSRleHBHNiMlInRHIiIiNyQiIiFGLUYrRisqKCUjYzFHRitGJ0YrNyQiIiRGK0YrRisqKEYvRitGJ0YrRixGK0YrKiYtRig2IywkRiohIiJGK0YsRitGKyooJSNjMkdGK0Y0Ris3JEYrRitGK0Y3KihGOUYrRjRGK0YsRitGKw==

evalm(%);

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evalm(A&*x);

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# Example 4x := c1*exp(t)*[-1,1,2] + c2*exp(2*t)*[-2,1,4] + c3*exp(3*t)*[-1,1,4];

NiM+JSJ4RywoKiglI2MxRyIiIi0lJGV4cEc2IyUidEdGKDclISIiRigiIiNGKEYoKiglI2MyR0YoLUYqNiMsJComRi9GKEYsRihGKEYoNyUhIiNGKCIiJUYoRigqKCUjYzNHRigtRio2IywkKiYiIiRGKEYsRihGKEYoNyVGLkYoRjhGKEYo

subs(t=0,%);

NiMsKCooJSNjMUciIiItJSRleHBHNiMiIiFGJjclISIiRiYiIiNGJkYmKiglI2MyR0YmRidGJjclISIjRiYiIiVGJkYmKiglI2MzR0YmRidGJjclRixGJkYyRiZGJg==

evalm(%) = [-1,0,0];

NiMvSyUndmVjdG9yRzYjNyUsKCUjYzFHISIiKiYiIiMiIiIlI2MyR0YtRiolI2MzR0YqLChGKUYtRi5GLUYvRi0sKComRixGLUYpRi1GLSomIiIlRi1GLkYtRi0qJkY0Ri1GL0YtRi1RKXBwcmludDQ3NiI3JUYqIiIhRjk=

B := matrix([[-1,-2,-1],[1,1,1],[2,4,4]]);

NiM+JSJCR0slJ21hdHJpeEc2IzclNyUhIiIhIiNGKjclIiIiRi1GLTclIiIjIiIlRjBRKXBwcmludDQ4NiI=

b := [-1,0,0];

NiM+JSJiRzclISIiIiIhRic=

# Use Maple's linsolve command to solve Bc=b:linsolve(B,b);

NiNLJSd2ZWN0b3JHNiM3JSIiISIiIiEiIlEpcHByaW50NDk2Ig==

subs({c1=0,c2=1,c3=-1},x); evalm(%);

NiMsKComLSUkZXhwRzYjJSJ0RyIiIjclIiIhRitGK0YpRikqJi1GJjYjLCQqJiIiI0YpRihGKUYpRik3JSEiI0YpIiIlRilGKSomLUYmNiMsJComIiIkRilGKEYpRilGKTclISIiRilGNEYpRjw=NiNLJSd2ZWN0b3JHNiM3JSwmKiYiIiMiIiItJSRleHBHNiMsJComRilGKiUidEdGKkYqRiohIiItRiw2IywkKiYiIiRGKkYwRipGKkYqLCZGK0YqRjJGMSwmKiYiIiVGKkYrRipGKiomRjpGKkYyRipGMVEpcHByaW50NTA2Ig==

# Example 5A := matrix([[1,-2,2],[-2,1,2],[2,2,1]]);

NiM+JSJBR0slJ21hdHJpeEc2IzclNyUiIiIhIiMiIiM3JUYrRipGLDclRixGLEYqUSlwcHJpbnQ1MTYi

evalm(A-r*I);

NiNLJSdtYXRyaXhHNiM3JTclLCYiIiJGKSUickchIiIhIiMiIiM3JUYsRihGLTclRi1GLUYoUSlwcHJpbnQ1MzYi

det(%)=0;

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factor(%);

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r1 := 3;

NiM+JSNyMUciIiQ=

u := vector(3);

NiM+JSJ1Ry0lJmFycmF5RzYkOyIiIiIiJDci

(A-r1*I)&*u;

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eq := evalm(%);

NiM+JSNlcUdLJSd2ZWN0b3JHNiM3JSwoKiYiIiMiIiImJSJ1RzYjRixGLCEiIiomRitGLCZGLjYjRitGLEYwKiZGK0YsJkYuNiMiIiRGLEYsRiksKComRitGLEYtRixGLComRitGLEYyRixGLComRitGLEY1RixGMFEpcHByaW50NTg2Ig==

eq1 := eq[1]=0; eq2 := eq[2]=0; eq3 := eq[3]=0;

NiM+JSRlcTFHLywoKiYiIiMiIiImJSJ1RzYjRilGKSEiIiomRihGKSZGKzYjRihGKUYtKiZGKEYpJkYrNiMiIiRGKUYpIiIhNiM+JSRlcTJHLywoKiYiIiMiIiImJSJ1RzYjRilGKSEiIiomRihGKSZGKzYjRihGKUYtKiZGKEYpJkYrNiMiIiRGKUYpIiIhNiM+JSRlcTNHLywoKiYiIiMiIiImJSJ1RzYjRilGKUYpKiZGKEYpJkYrNiNGKEYpRikqJkYoRikmRis2IyIiJEYpISIiIiIh

u[2] := v; u[3] := s;

NiM+JiUidUc2IyIiIyUidkc=NiM+JiUidUc2IyIiJCUic0c=

u[1] := solve(eq1,u[1]);

NiM+JiUidUc2IyIiIiwmJSJ2RyEiIiUic0dGJw==

evalm(u);

NiNLJSd2ZWN0b3JHNiM3JSwmJSJ2RyEiIiUic0ciIiJGKEYqUSlwcHJpbnQ2MTYi

u1 := evalm(subs({s=1,v=0},evalm(u)));

NiM+JSN1MUdLJSd2ZWN0b3JHNiM3JSIiIiIiIUYpUSlwcHJpbnQ2OTYi

u2 := evalm(subs({s=0,v=1},evalm(u)));

NiM+JSN1MkdLJSd2ZWN0b3JHNiM3JSEiIiIiIiIiIVEpcHByaW50NzA2Ig==

r3 := -3;

NiM+JSNyM0chIiQ=

u := vector(3);

NiM+JSJ1Ry0lJmFycmF5RzYkOyIiIiIiJDci

(A-r3*I)&*u;

NiMtJSMmKkc2JCwmJSJBRyIiIiomIiIkRiglIklHRihGKCUidUc=

eq := evalm(%);

NiM+JSNlcUdLJSd2ZWN0b3JHNiM3JSwoKiYiIiUiIiImJSJ1RzYjRixGLEYsKiYiIiNGLCZGLjYjRjFGLCEiIiomRjFGLCZGLjYjIiIkRixGLCwoKiZGMUYsRi1GLEY0KiZGK0YsRjJGLEYsKiZGMUYsRjZGLEYsLCgqJkYxRixGLUYsRiwqJkYxRixGMkYsRiwqJkYrRixGNkYsRixRKXBwcmludDczNiI=

eq1 := eq[1]=0; eq2 := eq[2]=0; eq3 := eq[3]=0;

NiM+JSRlcTFHLywoKiYiIiUiIiImJSJ1RzYjRilGKUYpKiYiIiNGKSZGKzYjRi5GKSEiIiomRi5GKSZGKzYjIiIkRilGKSIiIQ==NiM+JSRlcTJHLywoKiYiIiMiIiImJSJ1RzYjRilGKSEiIiomIiIlRikmRis2I0YoRilGKSomRihGKSZGKzYjIiIkRilGKSIiIQ==NiM+JSRlcTNHLywoKiYiIiMiIiImJSJ1RzYjRilGKUYpKiZGKEYpJkYrNiNGKEYpRikqJiIiJUYpJkYrNiMiIiRGKUYpIiIh

# Solve by Gaussian elimination:

eq2 := eq2 + eq1/2;

NiM+JSRlcTJHLywmKiYiIiQiIiImJSJ1RzYjIiIjRilGKSomRihGKSZGKzYjRihGKUYpIiIh

eq3 := eq3-eq1/2;

NiM+JSRlcTNHLywmKiYiIiQiIiImJSJ1RzYjIiIjRilGKSomRihGKSZGKzYjRihGKUYpIiIh

eq3 := eq3-eq2;

NiM+JSRlcTNHLyIiIUYm

eq1;eq2;eq3;

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u[3] := s;

NiM+JiUidUc2IyIiJCUic0c=

u[2] := solve(eq2,u[2]);

NiM+JiUidUc2IyIiIywkJSJzRyEiIg==

u[1] := solve(eq1,u[1]);

NiM+JiUidUc2IyIiIiwkJSJzRyEiIg==

evalm(u);

NiNLJSd2ZWN0b3JHNiM3JSwkJSJzRyEiIkYnRihRKXBwcmludDc0NiI=

u3 := evalm(subs(s=1,%));

NiM+JSN1M0dLJSd2ZWN0b3JHNiM3JSEiIkYpIiIiUSlwcHJpbnQ3NTYi

c1*exp(r1*t)*u1 + c2*exp(r2*t)*u2 + c3*exp(r3*t)*u3;

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x := evalm(%);

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# Check that dx/dt = Ax:evalm(map(diff,x,t));

NiNLJSd2ZWN0b3JHNiM3JSwoKigiIiQiIiIlI2MxR0YqLSUkZXhwRzYjLCQqJkYpRiolInRHRipGKkYqRioqKCIiI0YqJSNjMkdGKi1GLTYjLCQqJkYzRipGMUYqRipGKiEiIiooRilGKiUjYzNHRiotRi02IywkKiZGKUYqRjFGKkY5RipGKiwmKihGM0YqRjRGKkY1RipGKiooRilGKkY7RipGPEYqRiosJiooRilGKkYrRipGLEYqRioqKEYpRipGO0YqRjxGKkY5USlwcHJpbnQ3NzYi

evalm(A&*x);

NiNLJSd2ZWN0b3JHNiM3JSwoKigiIiQiIiIlI2MxR0YqLSUkZXhwRzYjLCQqJkYpRiolInRHRipGKkYqRioqKEYpRiolI2MyR0YqLUYtNiMsJComIiIjRipGMUYqRipGKiEiIiooRilGKiUjYzNHRiotRi02IywkKiZGKUYqRjFGKkY5RipGKiwmKihGKUYqRjNGKkY0RipGKiooRilGKkY7RipGPEYqRiosJiooRilGKkYrRipGLEYqRioqKEYpRipGO0YqRjxGKkY5USlwcHJpbnQ3ODYi

TTdSMApJNlJUQUJMRV9TQVZFLzEzNjUxNTI1MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyoiJCIkIiIiIiIhRihGKEYnRihGKEYoRic2Ig==TTdSMApJNlJUQUJMRV9TQVZFLzEzNTU5NzA1MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyoiJCIkLCQlInJHISIiIiIhRipGKkYnRipGKkYqCkYnNiI=