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We have the following linear matrix inequality (LMI) $$\begin{bmatrix} \mathrm A \,\, & \mathrm X\\ \mathrm X^{\top} & \mathrm B\end{bmatrix} \succeq \mathrm O$$ where $\mathrm X = \mathrm A^{\frac …

$$\mathrm y^\top \mbox{diag}(\mathrm A^\top \mathrm B \,\mathrm A) \,\mathrm y = \sum_{k=1}^n \mathrm e_k^\top\mathrm A^\top \mathrm B \,\mathrm A \,\mathrm e_k \, y_k^2 = \sum_{k=1}^n \mathrm a_k^\to …

We have the following system of quadratic equations in $\mathrm x \in \mathbb R^n$ $$\mathrm A (\mathrm x \circ \mathrm x) + \mathrm B \mathrm x + \mathrm c = 0_m$$ where $\mathrm A \in \mathbb R^{m …

H. Edwin Romeijn, Robert L. Smith, Shadow Prices in Infinite-Dimensional Linear Programming, Mathematics of Operations Research, Vol. 23, No. 1, February 1998. We consider the class of linear pr …

Let $\mathrm x, \mathrm y \in \mathbb R^n$. Let $\mathrm P_1$ and $\mathrm P_2$ be $n \times n$ permutation matrices such that the entries of $\mathrm P_1 \mathrm x$ and $\mathrm P_2 \mathrm y$ are in …

We have two coupled linear matrix equations in $\mathrm X_1, \mathrm X_2 \in \mathbb R^{n \times n}$ $$\begin{array}{rl} \alpha \, \mathrm A \mathrm X_1 + \mathrm X_1 \mathrm B &= \gamma \, \mathrm C …

Rephrasing: Given matrices $\mathrm A_1 \in \mathbb R^{m \times k}$ and $\mathrm A_2 \in \mathbb R^{k \times n}$, is there an upper bound on the number of nonzero entries of the $m \times n$ matri …

Let $\mathbb P_n$ be the set of $n \times n$ permutation matrices. Given matrix $\mathrm A \in \mathbb R^{m \times n}$ and vector $\mathrm v \in \mathbb R^n$, we would like to find a permutation matri …

Arguably, the simplest case is where the matrix $\bf A$ is symmetric and positive semidefinite (PSD) and ${\bf u} = {\bf v}$, which ensures that the eigenvalues of the rank-$1$ update are in $\Bbb R_0 …

To complement Christian's comment, since the spectral radius is upper-bounded by the spectral norm, $$\begin{aligned} \rho \left( {\bf A} + {\bf u} {\bf v}^\top \right) &\leq \left\| {\bf A} + {\bf u} …

The smallest eigenvalue can be found (approximately) via the following semidefinite program (SDP). $$ \begin{array}{ll} \underset {t} {\text{maximize}} & t \\ \text{subject to} & \operatorname{diag}({ …

Given a Hurwitz matrix $\mathrm A \in \mathbb R^{n \times n}$, let $$\Phi (t) := \exp(\mathrm A t)$$ be the state transition matrix, and let its $(i,j)$-th entry be denoted by $$\varphi_{ij} (t) := …

Minimizing $f$ and $h$ subject to the given constraints are quadratic programs of the form $$\begin{array}{ll} \text{minimize} & \frac 12 \mathrm x^{\top} \mathrm A \,\mathrm x\\ \text{subject to} & …

Given $\mathrm A \in \mathbb R^{n \times n}$, we define $f : \mathbb R \to \mathbb R$ as follows $$f (x) := \mbox{tr} \left( (\mathrm I_n + x \mathrm A)^{-1} (\mathrm I_n - x^2 \mathrm A) \right)$$ …

Suppose we are given $\mathrm M \in \mathbb R^{n \times n}$. Stating that all the eigenvalues of $\mathrm M$ have strictly negative real parts is equivalent to stating that there is a symmetric positi …