Test 3 practice exam

with(linalg):

I3:=diag(1,1,1);

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A:=matrix([[1,0,1],[0,2,0],[1,0,1]]);

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p:=det(evalm(A-lambda*I3));

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solve(p=0,lambda);

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v1:=nullspace(evalm(A));

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v2:=nullspace(evalm(A-2*I3));

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a. spectrum ={0,2} spectrial radius=2

b.{ \316\273=0,[-1,0,1]}, {\316\273=2,[1,0,1],[0,1,0]}

q1:=normalize(v1[1]);

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q2:=normalize(v2[1]);

PTYiNiM7IiIiIiIkRVxbbCRGJiwkKiQiIiMjRiZGK0YsRisiIiFGJ0Yp

q3:=normalize(v2[2]);

PTYiNiM7IiIiIiIkRVxbbCRGJiIiISIiI0YmRidGKQ==

Q:=augment(q1,q2,q3);

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evalm(transpose(Q)&*A&*Q);

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

PTYiNiQ7IiIiIiIjRiVFXFtsJTYkRidGJ0YnNiRGJkYmRic2JEYmRidGJjYkRidGJkYm

X:=matrix([[x],[y]]);

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simplify(evalm(transpose(X)&*A&*X));

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I2:=diag(1,1);

PUknc3BhcnNlRzYiNiQ7IiIiIiIjRiZFXFtsIzYkRihGKEYnNiRGJ0YnRic=

p:=det(evalm(A-lambda*I2));

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solve(p=0,lambda);

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b. both eigenvalues are postive.

c. 2>0 and det(A)=4-1=3 so postive definite.

gausselim(A);

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det(%);

Both piviots are postivie.

IiIk

p1:=nullspace(evalm(A-3*I2))[1];

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p2:=nullspace(evalm(A-I2))[1];

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H:=matrix([[1,1],[0,3]]);

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P:=det(evalm(H-lambda*I2));

KiYsJiIiIkYkSSdsYW1iZGFHNiIhIiJGJCwmIiIkRiRGJUYnRiQ=

p1a:=nullspace(evalm(H-3*I2))[1];

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p2a:=nullspace(evalm(H-I2))[1];

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For F. [1,1],[-1,1]; for H [1,2],[1,0]

Q:=augment(normalize(p1),normalize(p2));

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

F:=evalm(A);

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

Lambda:=evalm(transpose(Q)&*F&*Q);

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

P1:=augment(p1a,p2a);

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

Lambda:=evalm(inverse(P1)&*H&*P1);

P:=evalm(Q&*inverse(P1));

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

evalm(inverse(P)&*F&*P);

LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXJvd0dGJDYmLUknbXRhYmxlR0YkNjYtSSRtdHJHRiQ2Jy1JJG10ZEdGJDYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGQC8lK2dyb3VwYWxpZ25HRkAvJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGR0Y0Rj5GQUZDLUYyNictRjU2KC1GODYkUSIwRidGO0Y+RkFGQ0ZFRkgtRjU2KC1GODYkUSIzRidGO0Y+RkFGQ0ZFRkhGPkZBRkMvJSZhbGlnbkdRJWF4aXNGJy9GP1EpYmFzZWxpbmVGJy9GQlEmcmlnaHRGJy9GRFEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHUSV0cnVlRicvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zeby8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zpby8lJmZyYW1lR0Zpby8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHUSZmYWxzZUYnLyUtZXF1YWxjb2x1bW5zR0ZjcC8lLWRpc3BsYXlzdHlsZUdGY3AvJSVzaWRlR0Zmbi8lMG1pbmxhYmVsc3BhY2luZ0dGZm8vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRmNwRjtGOy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==

It is always good to check your answers.

A:=matrix([[2,3,1],[0,-1,1],[0,0,1]]);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkqbXZlcmJhdGltR0YkNiNRJSUiQUdGJy1JI21vR0YkNi1RIzo9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSShtZmVuY2VkR0YkNiYtRiM2Iy1JJ210YWJsZUdGJDY3LUkkbXRyR0YkNigtSSRtdGRHRiQ2KC1JI21uR0YkNiRRIjJGJ0YzLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zobi8lK2dyb3VwYWxpZ25HRmhuLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRl9vLUZWNigtRlk2JFEiM0YnRjNGZm5GaW5GW29GXW9GYG8tRlY2KC1GWTYkUSIxRidGM0ZmbkZpbkZbb0Zdb0Zgb0ZmbkZpbkZbby1GUzYoLUZWNigtRlk2JFEiMEYnRjNGZm5GaW5GW29GXW9GYG8tRlY2KC1GIzYkLUYwNi1RKiZ1bWludXMwO0YnRjNGNkY5RjtGPUY/RkFGQy9GRlEsMC4yMjIyMjIyZW1GJy9GSUZbcUZpb0ZmbkZpbkZbb0Zdb0Zgb0Znb0ZmbkZpbkZbby1GUzYoRl5wRl5wRmdvRmZuRmluRltvLyUmYWxpZ25HUSVheGlzRicvRmduUSliYXNlbGluZUYnL0ZqblEmcmlnaHRGJy9GXG9RJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR1EldHJ1ZUYnLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGXXIvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGaHIvJSZmcmFtZUdGaHIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0Y4LyUtZXF1YWxjb2x1bW5zR0Y4LyUtZGlzcGxheXN0eWxlR0Y4LyUlc2lkZUdGZXEvJTBtaW5sYWJlbHNwYWNpbmdHRmVyRjMvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=

Since an upper triangular then eigenvalues are on the diagonal and are 2,-1,1 so there are three linearly independent eigenvectors so it is diagonlizable.

B:=matrix([[2,1,-1],[1,-4,1],[-1,-1,9]]);

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1. circle centered at 2 radius 2, 2. circle centered at -4 radius 2, 3. circle centered at 9 radius 2.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUtSSNtb0dGJDYtUSJ+RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj5GNQ==in the interval [0,4], LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU= in the interval [-6,-2] and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU= in the interval [7,11] . There are three real eigenvalues so three distinct eigenvectors. This means that the matrix can be made similar to a diagonl matrix.

The interval for the largest eigenvlaue is [7,11]

C:=matrix([[7/8,1/8],[1/8,7/8]]);

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

x0:=matrix([[2],[1]]);

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

eigenvects(C);

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

Ther other eigenvalue is 3/4.

b. { 1,[1,1]}, {3/4,[-1,1]}

q1:=normalize([1,1]); q2:=normalize([-1,1]);

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PTYiNiM7IiIiIiIjRVxbbCNGJiwkKiYjRiZGJ0YmKUYnRitGJiEiIkYnLCRGKkYm

Q:=augment(q1,q2);

Lambda:=evalm(transpose(Q)&*C&*Q);

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

steadysc:=diag(1,0);

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

steadystate:=evalm(Q&*steadysc&*transpose(Q)&*x0);

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

B:=matrix([[1,3],[3,1]]);

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

simplify(evalm(transpose(X)&*B&*X));

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

EB:=eigenvects(B);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkqbXZlcmJhdGltR0YkNiNRJiUjRUJHRictSSNtb0dGJDYtUSM6PUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUYsNiNRXWl3NiQ3IzclLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYkLUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2LVEqJnVtaW51czA7NiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLyUmZmVuY2VHUSZmYWxzZTYiLyUqc2VwYXJhdG9yR1EmZmFsc2U2Ii8lKXN0cmV0Y2h5R1EmZmFsc2U2Ii8lKnN5bW1ldHJpY0dRJmZhbHNlNiIvJShsYXJnZW9wR1EmZmFsc2U2Ii8lLm1vdmFibGVsaW1pdHNHUSZmYWxzZTYiLyUnYWNjZW50R1EmZmFsc2U2Ii8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtNiIvJSdyc3BhY2VHUSwwLjIyMjIyMjJlbTYiLUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2JFEiMjYiLyUsbWF0aHZhcmlhbnRHUSdub3JtYWw2Ii1JI21uRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNiRRIjE2Ii8lLG1hdGh2YXJpYW50R1Enbm9ybWFsNiI8IzwjLUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYmLUknbXRhYmxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjUtSSRtdHJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2Jy1JJG10ZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYkLUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2LVEqJnVtaW51czA7NiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLyUmZmVuY2VHUSZmYWxzZTYiLyUqc2VwYXJhdG9yR1EmZmFsc2U2Ii8lKXN0cmV0Y2h5R1EmZmFsc2U2Ii8lKnN5bW1ldHJpY0dRJmZhbHNlNiIvJShsYXJnZW9wR1EmZmFsc2U2Ii8lLm1vdmFibGVsaW1pdHNHUSZmYWxzZTYiLyUnYWNjZW50R1EmZmFsc2U2Ii8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtNiIvJSdyc3BhY2VHUSwwLjIyMjIyMjJlbTYiLUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2JFEiMTYiLyUsbWF0aHZhcmlhbnRHUSdub3JtYWw2Ii8lKXJvd2FsaWduR1EhNiIvJSxjb2x1bW5hbGlnbkdRITYiLyUrZ3JvdXBhbGlnbkdRITYiLyUocm93c3BhbkdRIjE2Ii8lK2NvbHVtbnNwYW5HUSIxNiItSSRtdGRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2KC1JI21uRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNiRRIjE2Ii8lLG1hdGh2YXJpYW50R1Enbm9ybWFsNiIvJSlyb3dhbGlnbkdRITYiLyUsY29sdW1uYWxpZ25HUSE2Ii8lK2dyb3VwYWxpZ25HUSE2Ii8lKHJvd3NwYW5HUSIxNiIvJStjb2x1bW5zcGFuR1EiMTYiLyUpcm93YWxpZ25HUSE2Ii8lLGNvbHVtbmFsaWduR1EhNiIvJStncm91cGFsaWduR1EhNiIvJSZhbGlnbkdRJWF4aXM2Ii8lKXJvd2FsaWduR1EpYmFzZWxpbmU2Ii8lLGNvbHVtbmFsaWduR1EnY2VudGVyNiIvJStncm91cGFsaWduR1EnfGdyZnJsZWZ0fGdyaHI2Ii8lL2FsaWdubWVudHNjb3BlR1EldHJ1ZTYiLyUsY29sdW1ud2lkdGhHUSVhdXRvNiIvJSZ3aWR0aEdRJWF1dG82Ii8lK3Jvd3NwYWNpbmdHUSYxLjBleDYiLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtNiIvJSlyb3dsaW5lc0dRJW5vbmU2Ii8lLGNvbHVtbmxpbmVzR1Elbm9uZTYiLyUmZnJhbWVHUSVub25lNiIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbXxpcjAuNWV4NiIvJSplcXVhbHJvd3NHUSZmYWxzZTYiLyUtZXF1YWxjb2x1bW5zR1EmZmFsc2U2Ii8lLWRpc3BsYXlzdHlsZUdRJmZhbHNlNiIvJSVzaWRlR1EmcmlnaHQ2Ii8lMG1pbmxhYmVsc3BhY2luZ0dRJjAuOGVtNiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLyUlb3BlbkdRIls2Ii8lJmNsb3NlR1EiXTYiNyM3JS1JI21uRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNiRRIjQ2Ii8lLG1hdGh2YXJpYW50R1Enbm9ybWFsNiItSSNtbkc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYkUSIxNiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiPCM8Iy1JKG1mZW5jZWRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2Ji1JJ210YWJsZUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1LUkkbXRyRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNictSSRtdGRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2KC1JI21uRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNiRRIjE2Ii8lLG1hdGh2YXJpYW50R1Enbm9ybWFsNiIvJSlyb3dhbGlnbkdRITYiLyUsY29sdW1uYWxpZ25HUSE2Ii8lK2dyb3VwYWxpZ25HUSE2Ii8lKHJvd3NwYW5HUSIxNiIvJStjb2x1bW5zcGFuR1EiMTYiLUkkbXRkRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNigtSSNtbkc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjYkUSIxNiIvJSxtYXRodmFyaWFudEdRJ25vcm1hbDYiLyUpcm93YWxpZ25HUSE2Ii8lLGNvbHVtbmFsaWduR1EhNiIvJStncm91cGFsaWduR1EhNiIvJShyb3dzcGFuR1EiMTYiLyUrY29sdW1uc3BhbkdRIjE2Ii8lKXJvd2FsaWduR1EhNiIvJSxjb2x1bW5hbGlnbkdRITYiLyUrZ3JvdXBhbGlnbkdRITYiLyUmYWxpZ25HUSVheGlzNiIvJSlyb3dhbGlnbkdRKWJhc2VsaW5lNiIvJSxjb2x1bW5hbGlnbkdRJ2NlbnRlcjYiLyUrZ3JvdXBhbGlnbkdRJ3xncmZybGVmdHxncmhyNiIvJS9hbGlnbm1lbnRzY29wZUdRJXRydWU2Ii8lLGNvbHVtbndpZHRoR1ElYXV0bzYiLyUmd2lkdGhHUSVhdXRvNiIvJStyb3dzcGFjaW5nR1EmMS4wZXg2Ii8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbTYiLyUpcm93bGluZXNHUSVub25lNiIvJSxjb2x1bW5saW5lc0dRJW5vbmU2Ii8lJmZyYW1lR1Elbm9uZTYiLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW18aXIwLjVleDYiLyUqZXF1YWxyb3dzR1EmZmFsc2U2Ii8lLWVxdWFsY29sdW1uc0dRJmZhbHNlNiIvJS1kaXNwbGF5c3R5bGVHUSZmYWxzZTYiLyUlc2lkZUdRJnJpZ2h0NiIvJTBtaW5sYWJlbHNwYWNpbmdHUSYwLjhlbTYiLyUsbWF0aHZhcmlhbnRHUSdub3JtYWw2Ii8lJW9wZW5HUSJbNiIvJSZjbG9zZUdRIl02IkYn

b product of eigenvalues is -2*4 =-8 so we have a hyperbola

q1:=normalize(EB[2][3][1]);

q2:=normalize(EB[1][3][1]);

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

Q:=augment(q1,q2);

Y:=evalm([[u],[v]]);

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

Y=evalm(transpose(Q)&*X);

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

Lambda:=evalm(transpose(Q)&*B&*Q);

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

So the new equatyion will be

evalm(transpose(Y)&*Lambda&*Y)[1,1]=1;

LywmKiYiIiUiIiIpSSJ1RzYiIiIjRiZGJiomRipGJilJInZHRilGKkYmISIiRiY=

Angle of rotation is cos(\316\270)=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

L:=matrix([[1,1,1],[0,2,0],[1,0,1]]);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkqbXZlcmJhdGltR0YkNiNRJSUiTEdGJy1JI21vR0YkNi1RIzo9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSShtZmVuY2VkR0YkNiYtRiM2Iy1JJ210YWJsZUdGJDY3LUkkbXRyR0YkNigtSSRtdGRHRiQ2KC1JI21uR0YkNiRRIjFGJ0YzLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zobi8lK2dyb3VwYWxpZ25HRmhuLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRl9vRlVGVUZmbkZpbkZbby1GUzYoLUZWNigtRlk2JFEiMEYnRjNGZm5GaW5GW29GXW9GYG8tRlY2KC1GWTYkUSIyRidGM0ZmbkZpbkZbb0Zdb0Zgb0Zkb0ZmbkZpbkZbby1GUzYoRlVGZG9GVUZmbkZpbkZbby8lJmFsaWduR1ElYXhpc0YnL0ZnblEpYmFzZWxpbmVGJy9Gam5RJnJpZ2h0RicvRlxvUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdRJXRydWVGJy8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRl5xLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmlxLyUmZnJhbWVHRmlxLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGOC8lLWVxdWFsY29sdW1uc0dGOC8lLWRpc3BsYXlzdHlsZUdGOC8lJXNpZGVHRmZwLyUwbWlubGFiZWxzcGFjaW5nR0ZmcUYzLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYn

EL:=eigenvects(L);

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

a \316\273=2 a.m.=2 g.m.=1, \316\273=0, a.m =1 and g.m=1.

q1:=normalize(EL[1][3][1]);

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

q2:=normalize(EL[2][3][1]);

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

q3:=vector([0,1,0]);

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Q:=augment(q1,q2,q3);

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R:=evalm(transpose(Q)&*L&*Q);

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c. 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

a. T.

b. F A must be diagonalizable

c. F..

d. F must be greater than zero

e. T.

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