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### Is there a natural distance between skew hermitian matrices?

Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to ...

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### Eigenvector localizaiton

I have raised this sort of question before but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry ...

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### sharper interlacing

The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I ...

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### Generalizations of Oppenheim's inequality

The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...

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### S-matrix conjecture: status?

Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.

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### name for a matrix operation

If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?

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### matrices whose entries sum to zero

Let $A$ be a non-singular matrix and let $s(A)$ be the sum of its entries. Under which conditions can it be assured that $s(A) \neq 0$?
if you like, you can assume that $A$ is symmetric.
Here is an ...

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123 views

### eigenvector update formula

Suppose that $B$ is a Hermitian matrix with one known eigenpair $(\lambda,v)$. (assume its the smallest or largest pair, if you like). Form the rank one update $B+\rho bb^{T}$.
Now I'm interested in ...

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### 'Condition number' for Rayleigh-Ritz quotient

Suppose that $A$ is a Hermitian matrix and that $u,v$ are two vectors. Is there some known function $\kappa(A)$ so that $||u-v|| \leq \kappa(A) |\frac{u^{\*}Au}{u^{\*}u}-\frac{v^{\*}Av}{v^{\*}v}|$?
...

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### A question about matrices with more details

Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that
...

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### Simultaneous decomposition of three projectors

A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal ...

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### concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...

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### inverse-closed matrix spaces

Is there a known characterization of such spaces?
An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the ...

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### small sums of entries in submatrices - strange phenomenon

Suppose that $x \in \mathbb{R}^{n}$ is a vector of small positive fractions, i.e. $x_{i} \approx \frac{1}{n}$. The exact values are unknown. I form the matrix $M=diag(x)-\frac{xx^{T}}{2}$ which is a ...

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### Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello,
I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it.
I can ask it in two different ways. Perhaps depending on the reader, the ...