Questions tagged [matrices]
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.
3,317 questions
1
vote
1
answer
117
views
How to prove positive definiteness of a matrix under given premises?
${\bf A} \in {\Bbb R}^{n \times n}$ is a symmetric positive definite matrix, whose diagonal elements are all positive while off-diagonal elements are all non-positive. ${\bf U} \in {\Bbb R}^{n \times ...
- 75
44
votes
7
answers
2k
views
If $\det(M)=ab$ is it true that $M=AB$ with $\det(A)=a, \det(B)=b$?
For which (commutative) rings $R$ and dimensions $n$ is the following claim true (or false)?
Claim: For all $a,b$ any $n\times n$ matrix $M$ with coefficients in $R$ and $\det(M)=ab$, can be factored ...
- 5,805
2
votes
1
answer
165
views
A Loewner ordering problem
Let $A$ be a positive definite diagonal matrix, $B$ be a real matrix, and $C$ be a complex matrix. All are square matrices of dimension $n$. I am wondering if it's true that
$$\Big\|A+B^\top C^* C B\...
- 71
3
votes
1
answer
309
views
For which $k$ does a generic choice of $k$ $n \times n$ matrices span a subspace of $\mathrm{GL}(n)$?
If $\rho(n)$ are the Radon-Hurwitz numbers, then for $k \leq \rho(n)$ it is possible to find $k$ $n \times n$ real-valued matrices $A_1, \dots, A_k$ so that for any $(a_1,\dots,a_k) \neq 0$, the ...
- 507
-2
votes
0
answers
51
views
Eigenvalues of tridiagonal matrices with constant row and off-diagonal sums [closed]
Prove that the eigenvalues of tridiagonal matrix A are a subset of eigenvalues of tridiagonal matrix B.
Consider the sequence of real numbers $b_1, b_2, \ldots, b_m$. Compare the following two real ...
- 1
0
votes
2
answers
181
views
Elementwise unreachable matrix
Let $A \in \mathbb{F}^{n \times n}$ be a square matrix, and let $(i,j)$ denote the entry in the $i$-th row and $j$-th column of $A$.
We say that the position $(i,j)$ is unreachable if for all positive ...
- 11
-6
votes
1
answer
109
views
Determining if binary matrix with specific form has full rank [closed]
I have the following 15x15 binary matrix with a specific form:
$$\begin{bmatrix}
1&1&0&0&0&0&1&1&1&1&1&1&0&0&0 \\
1&0&1&0&0&...
- 1
13
votes
1
answer
364
views
Joint spectrum of two matrices and simultaneous upper triangulisation
I consider the following conjecture:
Let $A,B$ be $n\times n$ matrices over $\mathbb{C}$ (or any algebraically closed field of characteristic zero). The following are equivalent:
$\det(I+xA+yB)\in\...
- 882
0
votes
0
answers
90
views
Variant of Cordes Inequality
The classical Cordes inequality states the following: suppose that $\|\cdot\|$ is the usual matrix norm, $0 < \alpha \leq 1$, and $A, B$ are $n \times n$ positive semidefinite Hermitian matrices. ...
-1
votes
2
answers
115
views
Constructing an orthonormal set with given projections in a direct sum decomposition
Let $V$ be an $n$-dimensional real inner product space. Suppose we have $k\leq n/2$ orthonormal vectors $u_1, u_2, \dots, u_k \in V$.
Assume that there exist pairwise orthogonal subspaces $A,B,C \...
- 11
0
votes
0
answers
141
views
On tensor product and rank
I am getting confused by the tensor product. I would appreciate some basic insight.
I consider $M_2\otimes M_3$. (Here $M_n$ denotes the complex $n\times n$ matrices.) The dimension of this space is 4*...
- 109
0
votes
0
answers
130
views
Matrix factorizations under quotient ring
Let $(R, \mathfrak{m})$ be a regular complete local ring, let $f \in \mathfrak{m}^2$, and let $b \in \mathfrak{m}$ such that $b$ is a non zero divisor of both $R$ and $R/\langle f \rangle$. Denote $S=...
2
votes
0
answers
129
views
Lipschitz property of Frobenius norm of "standard deviation matrix"
Question: Is the Frobenius norm of (some form of) standard deviation matrix Lipschitz with respect to the Wasserstein distance?
To be more precise: Suppose $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ are two-...
- 21
2
votes
1
answer
191
views
Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that
$$
\underset{P^2 = P,\; \text{rank}(P) = p}{...
- 23
2
votes
0
answers
147
views
I am looking for "something like" an entry-wise matrix 1/2-norm. Has such a thing been studied? Where should I look?
Suppose an $n\times n$ square matrix $A$ with real or complex entries $A_{ij}$. Now define a quantity $Z(A)$ associated with the matrix by
$$
Z(A)=\sum_i \sum_j |A_{ij}|^{1/2}.
$$
What is this ...
