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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
    
@Stuart As the OP has approved your answer, please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. –  Julian Kuelshammer Jun 14 '13 at 7:26
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