[Math] Finding the eigenvalues of this 3×3 matrix

eigenvalues-eigenvectorslinear algebra

I entered this matrix into matlab and did the eig(a) function and I'm supposed to be getting roots -1, 1, and 3, but after working the problem out, I seem to be far from it.

Best Answer

At the 4th row of your working,

It is supposed to be (-7-$\lambda$)(-6+$\lambda$-6$\lambda$+$\lambda^2$-8) not (-7-$\lambda$)[-6+$\lambda$-6$\lambda$+$\lambda^2$+8).

Hope you can continue to work it from here.

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[Math] Eigenvalues of $A^T A$

A simple iterative algorithm to find the largest eigenvalue of a symmetric matrix $B$ is to compute repeatedly $x_{n+1} = B \, x_n$ starting from a random vector $x_0$, normalizing the length in each iteration. When it converges, you've found an eigenvector, and then you get the eigenvalue. There are some complications (repeated eigenvalues, event that $x_0$ is very near another eigenvector, etc), but that's the idea. Eg in Octave:

>>> A = rand(A,4); B = A'*A;  x = rand(1,4);
>>> x = x * B  /norm(x)
x =   1.7215   2.4537   2.3203   1.5036
>>> x = x * B  /norm(x)
x =   1.9544   3.0235   2.8072   1.6825
>>> x = x * B  /norm(x)
x =   1.9484   3.0362   2.8112   1.6717
>>> x = x * B  /norm(x)
x =   1.9477   3.0372   2.8113   1.6705
>>> x = x * B  /norm(x)
x =   1.9476   3.0373   2.8113   1.6704
>>> x = x * B  /norm(x)
x =   1.9476   3.0373   2.8113   1.6704
>>> norm(x * B) / norm(x)
ans =  4.8695
>>> eig(B)
ans =  0.074894   0.130166   0.485889   4.869534

[Math] How to solve the eigenvalues of a complex matrix of very high condition number

Don't know if this would work with a complex matrix, but partial pivoting may help out (LU factorization).

lu(A)

http://www.mathworks.com/help/matlab/ref/lu.html

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[Math] Eigenvalues of $A^T A$

[Math] How to solve the eigenvalues of a complex matrix of very high condition number

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