Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with matrix-theory
Search options not deleted
user 62673
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
12
votes
2
answers
791
views
A (linear) optimization problem subject to (linear) matrix inequality constraints
Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$ i …
CommunityBot
- 1
4
votes
3
answers
238
views
Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$
Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix
$$
X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}.
$$
Such matri …
- 1,769
1
vote
0
answers
131
views
Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\ …
- 2,712
16
votes
2
answers
502
views
The number of 0-1 normal matrices
Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$.
My question: What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, th …
- 11
4
votes
0
answers
248
views
Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite …
- 2,712
3
votes
0
answers
175
views
On a matrix inequality based on the Schur-Horn theorem
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.)
Let $A_s$ denote the symmetric part of $A …
- 2,712
1
vote
1
answer
326
views
On a condition for a matrix sum to be zero
Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite m …
- 1,517
2
votes
0
answers
248
views
A parametrization of stable matrices
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is pos …
- 2,712
1
vote
2
answers
1k
views
A "positive diagonal plus skew-symmetric" matrix decomposition
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric).
My question. Do there exist an orthogonal …
- 4,991
2
votes
0
answers
79
views
Characterizing a subclass of row-orthogonal matrices
Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,
$ …
- 2,712
2
votes
0
answers
54
views
Lower bounds on eigenvalues of Lyapunov solutions
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
$$
AX+XA^\top=-BB^\top.
$ …
- 2,712
1
vote
0
answers
172
views
A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive defi …
- 2,712
3
votes
1
answer
368
views
Closed-form expression for differential of matrix function
Let $X$ be a real $n\times n$ positive semidefinite matrix of rank $m\le n$ and let $Y\in\mathbb{R}^{m\times n}$ be the unique matrix satisfying (i) $X=Y^\top Y$, and (ii) $Y\, [I\, |\, 0]^\top = L$ w …
- 25.3k
3
votes
1
answer
1k
views
Completing the square of a matrix expression
Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ma …
- 20.2k
2
votes
1
answer
451
views
Constrained optimization over a trace functional
Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$
$$\begin{ar …
- 1,538