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Consider a $N \times N$ hermitian matrix $A$. Consider a complex $N \times 1$ vector $b$ and positive constant $c$. Given $A$ (hence its eigen-values), can we find the eigen-values of the following matrix. \begin{align} B=\begin{bmatrix} A & b \\ b' & c \end{bmatrix} \end{align}

in terms of those of $A$ (or $A$'s entries) and, $b$ and $c$.

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I think that you should add a question mark after "the matrix" – Belgi Oct 22 '12 at 13:20
@Belgi ok, I will do that – dineshdileep Oct 22 '12 at 13:22
What kind of characterization are you looking for? I don't believe there's a simple representation, otherwise this would be a really good way of computing eigenvalues! You can, however, say the sum of eigenvalues of B is the sum of eigenvalues of A plus c. – Stuart Nov 23 '12 at 7:20
I figured out that it isn't easy as it looks like later!!. And yes, your observation is really true. If it was true, one could start from a $2\times 2$ and thus find all the eigen-values of any $N\times N$ matrix :) , too good to be true! – dineshdileep Nov 23 '12 at 16:47

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