[This question](https://mathoverflow.net/questions/43514/how-do-eigenvectors-and-eigenvalues-change-when-we-remove-a-row-column-pair-of-a) asks about the changes in eigenvalues when a row and a column is added to a symmetric matrix. [Cauchy's interlacing theorem](http://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem) gives us a way to understand this case.


I am interested in knowing if there is any result or study for a real and non-symmetric matrix. That is, if $$B=\left[\begin{array}{ccc}
A & x\\\\
y & z \end{array}\right]$$ 


how do we relate the eigenvalues of $B$ to the eigenvalues of $A$ and the values in the vectors $x$ and $y%$ and the scalar $z$? Are there some bounds for the spectrum? 



How does appending a row and a column change eigenvalues of a non-symmetric matrix?
You need more than just $x$, $y$, $z$ and the eigenvalues of $A$ to determine or bound the eigenvalues of $B$. For example, consider
$$ B = \pmatrix{1 & 0 & | & 1\cr t & 0 & | & 0\cr \hline 0 & 1 & | & 0\cr}$$
where $A$ has eigenvalues $1$ and $0$, but any real number $\lambda$ will be an eigenvalue of $B$ for suitable $t$ (namely $t = \lambda^3 - \lambda^2$).


If $S$ diagonalizes $A$ so that $S^{-1} A S = \Lambda$ is diagonal, then
$$ \pmatrix{S & 0\cr 0 & 1}^{-1} B \pmatrix{S & 0 \cr 0 & 1\cr} = \pmatrix{\Lambda & S^{-1}x\cr y S & z\cr} $$
so by Gerschgorin's theorem, the eigenvalues of $B$ are contained in the union of the disks
centred at $\lambda\_j$ of radii $\left|(S^{-1} x)\_j\right|$ and the disk centred at $z$ of
radius $\left\|yS\right\|\_1$. 


