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I am wondering if there is something known about the interlacing properties of an "Almost Hermitian" matrix, in the following sense: let A be a nxn matrix so that it has a Hermitian principal minor of order (n-1)x(n-1). What interlacing properties does A possess?


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Assuming that the Hermitian minor has distinct eigenvalues, there is nothing we can say about the eigenvalues of $A$. Let the eigenvalues of the Hermitian minor be $\lambda_1$, ..., $\lambda_{n-1}$, so we can choose a matrix where your matrix looks like $$A=\begin{pmatrix} \lambda_1 & & & & a_1 \\ & \lambda_2 & & & a_2 \\ & & \lambda_3 & & a_3 \\ & & & \ddots & & \\ b_1 & b_2 & b_3 & & c \end{pmatrix}$$

The characteristic polynomial of $A$ is $$f(x):=(x-c) \prod_{i} (x-\lambda_i) - \sum_j a_j b_j \prod_{i \neq j} (x-\lambda_i). \quad (\ast)$$ I claim that we can choose $c$ and $a_j b_j$ to make $f(x)$ be any monic degree $n$ polynomial.

Proof: We have $f(\lambda_j) = - a_j b_j \prod_{i \neq j} (\lambda_i - \lambda_j)$. So, assuming that the $\lambda$'s are distinct, we can choose $a_j b_j$ to make $f(\lambda_j)$ have any value. Also, we can use $c$ to fix the value of $f$ at any $x$ other than the $\lambda_i$. A monic polynomial of degree $n$ is determined by its values at $n$ points, so we can arrange for $f$ to be any degree $n$ monic polynomial. $\square$

Equation $(\ast)$ also shows that, if $\lambda_i$ occurs with multiplicity $d$ in the Hermitian minor, then it occurs with multiplicity $\geq d-1$ in $A$.

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Thanks! Follow-up question pops up, though: what if I add the extra constraint that the c and b vectors are multiples of each other? Thanks – Felix Goldberg Mar 27 '12 at 19:55

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