Modified skew-symmetric matrix eigenvalues

Question

I know that the eigenvalues of a real skew-symmetric matrix $A$ are either zero or imaginary. I have the following modification: let $C$ be a diagonal matrix with ones for some diagonal elements and zeroes for others. Can I say anything about the eigenvalues of the matrix $B=(C+(I-C)A(I-C))$? This is a matrix where the column and rows corresponding to $C_{ii}=1$ have been zeroed out and the diagonal element has been set to one: $B_{ii} = 1$.

Answer 1

If you rearrange the rows and columns so that all the one entries of $C$ are at the top (i.e., conjugate by a permutation, which preserves skew-symmetry of $A$), then there is a block diagonal format where we can express

$$ C = \begin{pmatrix} I & 0 \\ 0 & 0 \end{pmatrix}$$

$$ A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22}\end{pmatrix}$$

so that we have

$$ B = \begin{pmatrix} I & 0 \\ 0 & A_{22}\end{pmatrix}$$

Since $A$ is skew-symmetric, we have $A_{21}=-A_{12}^T$ and $A_{11}=-A_{11}^T$ and $A_{22}=-A_{22}^T$.

The spectrum of $B$ will be the spectrum of $I$ union the spectrum of $A_{22}$, and since $A_{22}$ is skew symmetric its eigenvalues will be $0$ or imaginary.

The spectrum of $A_{22}$ won't necessarily be a subset of the spectrum of $A$; consider $C=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. The spectrum of $A$ is $\pm i$ but $A_{22}=0$. I don't know what kind of relationship exists between the spectrum of $A_{22}$ and $A$, if that's what you are asking.