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let me assume $A$ is invertible, then you ask when $$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$ so if $X$ has eigenvalues $x_i$, $i=1,2,\ldots n$, you would need $$\prod_{i}(1+x_i)=1+\prod_i x_i$$ basically …

This is a so-called chiral symmetry. The restriction on the symmetry of the spectrum of $M$ is the only restriction you need, you can then bring $M$ to the desired off-diagonal form by a unitary trans …

To see what you might expect for a relation, consider the case of a $2\times 2$ matrix $M=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, with eigenvalues $\lambda_\pm=\tfrac{1}{2}(a+d)\pm\sqrt{4bc+(a-d)^2}$. …

Consider a $2n\times 2n$ real matrix $M$ that satisfies the pseudo-unitarity condition $$M^{\rm T}\Lambda M=\Lambda,\;\;\Lambda=\begin{pmatrix}1_n & 0\\ 0 & -1_n\end{pmatrix}.\qquad [1]$$ (The matri …

On efficient sparse integer matrix Smith normal form computations (Dumas, Saunders, Villard, 2001) We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. W …

this is not a complete answer, but it's a bit too long for a comment. first notice that ${\rm Det}\;(\lambda-M)={\rm Det}\;(-\lambda-M)$ and ${\rm Det}\;(\lambda-M)={\rm Det}\;(\bar{\lambda}-M)$; it …

If you decompose $M=\begin{pmatrix} X_{q\times q}&Y_{q\times k_3}\\ (Y_{q\times k_3})^{\rm T}&0_{k_3\times k_3}\end{pmatrix}$ into four block matrices, with $q=k_1+k_2$, then the determinant equals $$ …

A closed form expression for finite $N$ is most certainly not available. A large-$N$ calculation for Gaussian distributed matrix elements has been reported in Density of eigenvalues of random band ma …