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I have this matrix

\begin{bmatrix}1&7&3\\7&4&-5\\3&-5&6\end{bmatrix}

I found that its eigenvalues are :

-7.008, 10.972, 7.036

when I use these eigenvalues to find eigenvectors by using

(A-λI)X = 0

where X is the eigenvector, I reduce the matrix (A-λI) to the RRE form, and I always end up with Identity matrix or

\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}

which results in X being

\begin{bmatrix}0\\0\\0\end{bmatrix}

which is not possible, so I tried using Wolfram Alpha and it did give me vectors when I saw the solution steps the program multiplied both sides of the equation by a matrix.

This the solution for λ=7.03595

what is the matrix and where did it come from?

I tried using Symbolab but it said that there are no eigenvectors for λ=-7.00793 and λ=10.97198

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    $\begingroup$ @Moo alright I will rewrite my method of reducing the matrix and post it $\endgroup$ – AdoobII Oct 22 '17 at 16:58
  • $\begingroup$ I used bluebit.gr/matrix-calculator/calculate.aspx and when I entered your matrix, it gave me nonzero vectors. The eigenvalues are "awful" so I suspect that if you want to do this algebraically, it won't look nice at all $\endgroup$ – imranfat Oct 24 '17 at 17:15

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