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Some call them currency exchange matrices. From Boyd & Vandenberghe's Introduction to Applied Linear Algebra: 6.7 Currency exchange matrix. We consider a set of $n$ currencies, labeled $1,\dots,n$. ( …

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}({ …

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} …

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 …

Rephrasing slightly, given (symmetric) matrix $\mathrm A \succeq \mathrm O_n$, we have the following minimization problem in (symmetric) matrix $\mathrm X \succeq \mathrm O_n$ $$\begin{array}{ll} \te …

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 …

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 …

Let $n \times n$ matrix $\rm A$ be symmetric and positive definite. Since $\rm A$ is symmetric, it is diagonalizable. Hence, there exists a (non-singular) matrix $\rm P$ such that $\mathrm A = \mathrm …

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 …

$$\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 …

The set of $n \times n$ symmetric Toeplitz matrices is $$\left\{ x_1 \mathrm M_1 + x_2 \mathrm M_2 + \cdots + x_n \mathrm M_n \mid x_1, x_2, \dots, x_n \in \mathbb R \right\}$$ where $\mathrm M_1, \m …

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 …

Given $\mathrm A \in \mathbb R^{m \times n}$ and $\mathrm B \in \mathbb R^{n \times p}$, let $\mathrm B \mathrm B^{\top} = \mathrm Q \Lambda \mathrm Q^{\top}$ be an eigendecomposition of $\mathrm B \m …

Let function $f : \mathbb R^{m \times n} \to \mathbb R_0^+$ be defined as follows $$f (\mathrm X) := \| \,\mathrm X \mathrm A^\top \|_\text{F}^2 + \| \,\mathrm X^\top \mathrm A \,\|_\text{F}^2 + \lef …

$$\mathrm A \mathrm x = \begin{bmatrix} \mathrm A_1 & \mathrm A_2 & \cdots & \mathrm A_n\end{bmatrix} \begin{bmatrix} \mathrm x_1\\ \mathrm x_2\\ \vdots \\ \mathrm x_n\end{bmatrix}$$ where $\mathrm x …