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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.
9
votes
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
Note that you are considering currency exchange matrices where $R_{ij} R_{ji} = 1$, which one could call perfect currency exchange matrices. … I would have written $R_{ij} R_{ji} \leq 1$ instead, in order to include the (reciprocal) perfect currency exchange matrices, too. …
0
votes
Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf ...
The smallest eigenvalue can be found (approximately) via the following semidefinite program (SDP).
$$ \begin{array}{ll} \underset {t} {\text{maximize}} & t \\ \text{subject to} & \operatorname{diag}({ …
1
vote
Spectral radius of a rank-1 perturbation
To complement Christian's comment, since the spectral radius is upper-bounded by the spectral norm,
$$\begin{aligned} \rho \left( {\bf A} + {\bf u} {\bf v}^\top \right) &\leq \left\| {\bf A} + {\bf u} …
0
votes
Spectral radius of a rank-1 perturbation
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 …
2
votes
Accepted
Roots of determinant of matrix with polynomial entries
Let $r_i := p_i - q_i$.
$${\bf A} (x) := \begin{bmatrix} p_1 (x) & q_1 (x) & \ldots & q_1 (x)\\ q_2 (x) & p_2 (x) & \ldots & q_2 (x)\\ \vdots & \vdots & \ddots & \vdots\\ q_n (x) & q_n (x) & \ldots & …
3
votes
Symmetric linear least-squares solution
Complementing Denis Serre's answer and rephrasing the original problem slightly, given tall matrices $\rm A$ and $\rm B$, we have the following quadratic program in square matrix $\rm X$
$$\begin{array …
6
votes
Accepted
Minimization problem involving the inverse of an affine matrix function
Rephrasing slightly, given (symmetric) matrix $\mathrm A \succeq \mathrm O_n$, we have the following minimization problem in (symmetric) matrix $\mathrm X \succeq \mathrm O_n$
$$\begin{array}{ll} \te …
2
votes
Calculate percentage of symmetry of a given matrix
Consider the following $2 \times 2$ matrix and its decomposition in a "natural" orthonormal basis.
$$\begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix} + \begin{b …
5
votes
Accepted
Positive definite matrices diagonalised by orthogonal matrices that are also involutions
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 …
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 …
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
$$x \mapsto \begin{cases} 1 & \text{if } x \neq 0\\ 0 & \text …
1
vote
Accepted
How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently?
$$\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 …
7
votes
Accepted
Finding Toeplitz matrix nearest to a given matrix
M_2, \dots, \mathrm M_n$ are $n \times n$ symmetric Toeplitz basis matrices. … Let $\mathrm M_1 = \mathrm I_n$ correspond to the main diagonal, whereas the remaining basis matrices correspond to super and sub diagonals. …
1
vote
Coupled Sylvester equations
We have two coupled linear matrix equations in $\mathrm X_1, \mathrm X_2 \in \mathbb R^{n \times n}$
$$\begin{array}{rl} \alpha \, \mathrm A \mathrm X_1 + \mathrm X_1 \mathrm B &= \gamma \, \mathrm C …
2
votes
Nearest matrix orthogonally similar to a given matrix
X \in \mathbb R^{n \times n}$
$$\begin{array}{ll} \text{minimize} & \| \mathrm A \mathrm X - \mathrm X \mathrm B \|_2\\ \text{subject to} & \mathrm X^\top \mathrm X = \mathrm I_n\end{array}$$
where matrices …