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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"2012-10-22
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    @Belgi ok, I will do that2012-10-22
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    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.2012-11-23
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    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!2012-11-23
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    @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](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-14

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