I do not understand a really basic part of Lemma 1 in this arxiv paper. For completeness, the lemma is:
Let one eigenvalue of $A$ be zero, WLOG we can set $\lambda_n(A) = 0$. Then,
$$\prod_{i=1}^{n-1} \lambda_i(A) |\det(B v_n)^2| = \det(B^* A B) $$
for any $n \times n – 1$ matrix $B$.
Above, $A$ is an $n \times n$ Hermitian matrix with $\lambda_i(A)$ eigenvalues and $v_i$ eigenvectors. My question is: if $B$ is a $n \times n – 1$ matrix and $v_n$ is an $n$-vector, how is $B v_n$ a valid matrix-vector multiplication?
In the paper, there's a weird space between $B$ and $v_n$, and I'm not sure if I'm missing something obvious.
Best Answer
I emailed the first author, and he said that "$B\;\;\ v_n$" is not a matrix-vector multiplication, but an $n \times n$ matrix or
$$ \begin{bmatrix} B & v_n \end{bmatrix} $$
This explains the weird spacing.