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Well, May be my question does not make any sense, But one of my junior asked me whether we can say eigenvalues of a matrix by inspection may be for $3\times 3$ matrix? He said for the following matrix, will $3$ be an eigenvalue with multiplicity $3$ with out calculation? $$\left( \begin{array}{ccc} 3 & 2 & 2 \\ 2 & 3 & 2 \\ 2 & 2 & 3 \\ \end{array} \right).$$ So far I know that if it is real symmetric then its eigenvalues are purely real, if skew then purely imaginary, if diagonal then all the entries, and if unitary then with modulus 1, known facts at all.If any one knows more about this please write.

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The eigenvalues of your matrix are $1$ (multiplicity $2$) and $7$. –  copper.hat Jun 2 '12 at 9:22
    
So, I hope there is no magic to guess eigen values by inspection right in general? –  Bunuelian Trick Jun 2 '12 at 9:23
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No more likely than solving a cubic equation by inspection. –  copper.hat Jun 2 '12 at 9:24
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2 Answers 2

up vote 4 down vote accepted

In this case you can see the eigenvalues "by inspection".

Item #1: If you subtract the identity matrix from your matrix, you get three repeated rows, i.e. $A-I$ has rank $1$ only. This means that the eigenspace associated with $\lambda=1$ is $2$-dimensional, IOW it is a double eigenvalue (and a double root of characteristic polynomial).

Item #2: The sum of entries on all rows is equal to $7$. This means that the vector $(1,1,1)^T$ is an eigenvector belonging to $\lambda=7$.

This is a 3x3 matrix, so that's all.

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The Gershgorin circle theorem comes close to estimating the eigenvalues by 'inspection' - by summing the absolute values of the row elements (except the ones on the diagonal). So in this case the three eigenvalues are all in the interval [3-4,3+4].

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Thanks for posting this. I was not previously aware of the Gershgorin circle theorem. and I'm glad you brought it to my attention. I'm starting to study spectral graph theory, and it seems to me that this theorem will be particularly helpful for understanding the eigenvalues of graph adjacency matrices, which have small row sums and always have zeroes on the diagonal. –  MJD Jun 3 '12 at 4:41
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