# Tagged Questions

The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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### Bounds on Matrix Exponential

Suppose $A$ and $B$ are (non-commuting) hermitian $n\times n$ matrices and $k$ is a large positive number. Suppose we write the product of matrix exponentials as $e^{kA + B} e^{-kA} = e^{C(k)}$ for ...
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### Eigenvalues of partial Hankel matrices

I was wondering if there are closed formulas for the singularvalues of a partial Hankel matrix (by partial I mean $\ell<n$) \begin{align*} H= \begin{bmatrix} c_1 & c_2 & \ldots & c_\...
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### Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$. My question is the following: Does it always exists a ...
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### Minimizing the operator norm of a sum of matrices

Given an $m\times n$ real matrix $C$. Let $a_i\in\{-1,1\}$ and $b_i\in\{-1,1\}$. Consider $$q=\left\|\text{diag}(a_1,\ldots,a_m)C + C\text{diag}(b_1,\ldots,b_n)\right\|$$ where the norm is the ...
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### When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?
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### approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
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### A question on Perron–Frobenius theorem [closed]

Let $A \in M_n$ is nonnegative(all $a_{ij}\ge0$). Suppose $A$ has a nonnegative eigenvector(all entries$\ge0$ ) with $r ≥ 1$ positive entries and $n − r$ zero entries. Why is there a permutation ...
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### Efficient SVD of a matrix without some of the columns

I have a matrix $A \in \mathbb{R}^{p \times q}$ of rank $r$ and its SVD decomposition, i.e, $$A = U S V^\top,$$ where $U \in \mathbb{R}^{p \times r}$ and $V \in \mathbb{R}^{q \times r}$ are ...
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### On the positive definiteness of RBF kernel with DTW distance

Consider an RBF kernel that is defined as $K(\mathbf{x},\mathbf{y}) = \exp\left(-\dfrac{d^2(\mathbf{x},\mathbf{y})}{2\sigma^2}\right)$, where $d(\mathbf{x},\mathbf{y})$ is usually chosen as the ...
The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite. $\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal. and the ...