This question asks about the changes in eigenvalues when a row and a column is added to a symmetric matrix. Cauchy's 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?