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Let $A_n\in\mathbb{R}^{n\times n}$ be defined as $$ A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 & b & a & \cdots & \cdots & 0 & 0\\ \vdots & \vdot …

Consider two measurable spaces $X_1 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_1)$ and $X_2 = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_2)$ and the product spaces $$X_1^{q} = (\times_{i=1}^q\math …

Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$. Question: Are there any upper bounds on the condition number of the eigenvector …

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation $$ AX+XA^\top=-BB^\top. $ …

Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and sufficie …

Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$. My question. For any fixed $i\in\{1,\dots,n\}$, …

Consider the linear time invariant system: $$\tag{1}\label{eq1} \dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n, $$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s …

Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration \begin{align} x_{k+1} = f(x_k,\ …