Finding Eigenvalues and Eigenvectors with Maple restart; with(LinearAlgebra): Consider the following matrix. A := Matrix([[34, -9, 81], [24, -8, 54], [-12, 3, -29]]);

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First, we find the characteristic polynomial p := CharacteristicPolynomial(A, lambda);

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Then we find the roots of p. factor(p);

KiYsJkknbGFtYmRhRzYiIiIiISIiRiZGJiwmRiRGJiIiI0YmRik=

solve(p=0, lambda);

NiUiIiIhIiNGJA==

First consider the Eigenvalue 1. The matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GU0Y1LUYsNiVRKCYjOTU1O0lGJ0YvRjJGOQ== is B := A - 1;

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R := ReducedRowEchelonForm(B);

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We can get Maple to solve the homogenous system LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ4RicvJSVib2xkR0YxRi8vRjNRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWLUkjbW5HRiQ2JlEiMEYnRjgvRjNGPkY8RkM= by backward substitution. The "free" option says which letter to use for the free variables. BackwardSubstitute(R, <0,0,0>, free=t);

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We can factor this by eye as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== times the vector v1 := <-3,-2,1>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSN2MUdGKC1JJ1JUQUJMRUdGKDYlIihnIm8qKS1JJ01BVFJJWEdGKDYjNyU3IyEiJDcjISIjNyMiIiImSSdWZWN0b3JHRiU2I0knY29sdW1uR0YoNyMtRjw2Iy9JJCVpZEdGKEYx

So, the eigenspace for 1 is one dimensional with v1 as a basis. Next, consider the eigenvalue -2 B := A - (-2);

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R := ReducedRowEchelonForm(B);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJSR0YoLUknUlRBQkxFR0YoNiUiKSEzIno2LUknTUFUUklYR0YoNiM3JTclIiIiIyEiIiIiJSMiIipGOjclIiIhRj5GPkY9SSdNYXRyaXhHRiU3Iy1GPzYjL0kkJWlkR0YoRjE=

BackwardSubstitute(R, <0,0,0>, free=t);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIoO0kyKi1JJ01BVFJJWEdGKDYjNyU3IywmJkkidEdGKDYjIiIjIyIiIiIiJSZGNzYjRjsjISIqRjw3I0Y2NyNGPSZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig3Iy1GQzYjL0kkJWlkR0YoRi8=

Factor by eye u1 := <-9/4, 0, 1>; u2 := <1/4, 1, 0>;

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Check that we did it right. t[1]*u1 + t[2]*u2;

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Check with maple's NullSpace command. NullSpace(R);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8JC1JJ1JUQUJMRUdGKDYlIig/XEgoLUknTUFUUklYR0YoNiM3JTcjIyIiIiIiJTcjRjc3IyIiISZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRigtRi42JSIoISlbSSktRjI2IzclNyMjISIqRjhGOkY5Rjw3IzwkLUY8NiMvSSQlaWRHRihGMC1GPDYjL0ZORkI=

Thus, the eigenspace for -2 is two dimensional with basis u1, u2. Now we want to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and a diagonal matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 so that 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. (D is reserved in Maple.) P := <v1| u1| u2>;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJQR0YoLUknUlRBQkxFR0YoNiUiKWshb0kiLUknTUFUUklYR0YoNiM3JTclISIkIyEiKiIiJSMiIiJGOjclISIjIiIhRjw3JUY8RjxGP0knTWF0cml4R0YlNyMtRkE2Iy9JJCVpZEdGKEYx

M := DiagonalMatrix([1, -2, -2]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJNR0YoLUknUlRBQkxFR0YoNiUiKCdwZyoqLUknTUFUUklYR0YoNiM3JTclIiIiIiIhRjg3JUY4ISIjRjg3JUY4RjhGOkknTWF0cml4R0YlNyMtRjw2Iy9JJCVpZEdGKEYx

P^(-1).A.P;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIpIW9vZyItSSdNQVRSSVhHRig2IzclNyUiIiIiIiFGNjclRjYhIiNGNjclRjZGNkY4SSdNYXRyaXhHRiU3Iy1GOjYjL0kkJWlkR0YoRi8=

Let's check ourselves using the Maple Eigenvectors command. See the help page for the output format. Eigenvectors(A);

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