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Results tagged with matrices
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user 91764
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
2
votes
Calculate percentage of symmetry of a given matrix
Consider the following $2 \times 2$ matrix and its decomposition in a "natural" orthonormal basis.
$$\begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix} + \begin{b …
1
vote
Using iterative projection to solve a minimization problem
Given matrices $\mathrm X_0, \mathrm C \in \mathbb R^{n \times n}$,
$$\begin{array}{ll} \text{minimize} & \| \mathrm X - \mathrm X_0 \|_{\text{F}}^2\\ \text{subject to} & \mathrm X 1_n = 1_n\\ & 1_n^{ …
1
vote
Accepted
Bounding the norm of a contraction matrix
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 …
1
vote
Accepted
Common lower bounds for positive semidefinite matrices
We have $n+1$ linear matrix inequalities (LMIs) in $\mathrm X$, namely,
$$\mathrm X \succeq \mathrm O_m, \qquad \mathrm P_1 - \mathrm X \succeq \mathrm O_m, \qquad \mathrm P_2 - \mathrm X \succeq \ma …
1
vote
Accepted
How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently?
$$\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 …
1
vote
Accepted
Maximise singular value decay by sparse matrix approximation
Maximizing the "decay" of the singular values could be thought of as minimizing the (numerical) rank. Hence, I believe that the original problem could be rephrased as follows:
Given $\mathrm A \in …
5
votes
Large power of an adjacency matrix
Since the adjacency matrix $\mathrm A$ is not symmetric, we have a directed graph.
Given a positive integer $k$, the $(i,j)$-th entry of $\mathrm A^k$ gives us the number of directed walks of lengt …
1
vote
Minimizing $\|X-X_0\|_F^2+\|X^{-1}-Y_0\|^2_F$ with respect to matrix $X$
Given $\mathrm X_0, \mathrm Y_0 \in \mathbb R^{n \times n}$,
$$\begin{array}{ll} \text{minimize} & \|\mathrm X - \mathrm X_0\|_F^2 + \|\mathrm Y - \mathrm Y_0\|_F^2\\ \text{subject to} & \mathrm Y = …
1
vote
Minimizing $\|X-X_0\|_F^2+\|X^{-1}-Y_0\|^2_F$ with respect to matrix $X$
Given $\mathrm X_0, \mathrm Y_0 \in \mathbb R^{n \times n}$,
$$\begin{array}{ll} \text{minimize} & \|\mathrm X - \mathrm X_0\|_F^2 + \|\mathrm Y - \mathrm Y_0\|_F^2\\ \text{subject to} & \mathrm Y \m …
5
votes
Do these matrices have a name?
the Sylvester construction
$$\mathrm H_{2k} = \begin{bmatrix} \mathrm H_k & \mathrm H_k\\ \mathrm H_k & -\mathrm H_k\end{bmatrix} \qquad \qquad \qquad \mathrm H_1 = 1$$
which builds (symmetric) Walsh matrices …
0
votes
Sparse matrix approximation using only a few dense columns (or rows)
Selecting $s \leq n$ rows can be done by left-multiplying $\rm A$ by a Boolean diagonal matrix with $s$ ones on its main diagonal. Hence, we have the following Boolean optimization problem
$$\begin{a …
1
vote
Inequality on diagonal entries of a matrix product
Let $\rm A, B, D$ be $n \times n$ matrices. …
1
vote
Coupled Sylvester equations
We have two coupled linear matrix equations in $\mathrm X_1, \mathrm X_2 \in \mathbb R^{n \times n}$
$$\begin{array}{rl} \alpha \, \mathrm A \mathrm X_1 + \mathrm X_1 \mathrm B &= \gamma \, \mathrm C …
1
vote
Accepted
Upper bound on the number of non-zero entries of the product of sparse matrices
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 $m \times n$ matrix … Let Boolean matrices $\mathrm B_1 \in \{0,1\}^{m \times k}$ and $\mathrm B_2 \in \{0,1\}^{k \times n}$ be obtained by applying the function
$$x \mapsto \begin{cases} 1 & \text{if } x \neq 0\\ 0 & \text …
7
votes
Accepted
What is the term for this type of matrix?
When $b = 0$, we have an $n \times n$ symmetric arrowhead matrix. When $b \neq 0$, we have
$$\begin{bmatrix}
c & c & c & \cdots & c & c \\
c & a & b & \cdots & b & b \\
c & b & a & \cdots & b & b \\
…