Question

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?

Answer 1

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 $\left\|yS\right\|_1$.