1
vote
1answer
102 views

Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let \begin{equation*} p = p\left(\kappa,d\right) := ...
4
votes
0answers
47 views

Homogeneity degree one functions of a matrix argument

I am interested in homogeneity degree one (scalar-valued) functions of a matrix argument. The simplest setup is as follows. Let $X$ be a symmetric $3\times 3$ matrix with real entries. Let $f$ be a ...
-1
votes
1answer
446 views

existence of polynomial equation system solution

For $1 \leq i \leq n$, let $A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \\ \end{bmatrix}$ $B_i=\begin{bmatrix} b_{i1} ...