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.
3,325 questions
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Decomposition $PLUP^{−1}$
This is a cross-post as I didn't get any answer. Let $A\in\mathrm{M}_n(\mathbb{C})$. If the leading principal minors (namely the determinants of the top left submatrices) of $A$ are non-zero, then ...
- 605
6
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0
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286
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Invertible perturbations of matrices
Let $P \in M_n(\mathbb{C})$ be a rank $k$ orthogonal projection, $k \geq 2$, and let $A_1, \ldots, A_r \in M_n(\mathbb{C})$ be matrices. Suppose that for every rank $k - 1$ orthogonal projection $Q &...
- 44.3k
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98
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(Weak) Jordan-Chevalley decompositions over non-perfect fields
This is a reference/literature request.
Given a field $K$ and an endomorphism $x \colon V \to V$ of a finite-dimensional $K$-vector space it is well-known that the Jordan-Chevalley decomposition of ...
- 31
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1
answer
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Cayley transform - determinant/permanent faster computation
Let $A$ be a $0/1$ matrix in $\mathbb Z^{n\times n}$ such that $I+A$ is invertible $\bmod 3$. Consider $Q=(I-A)(I+A)^{-1}$.
Let $Det(M)$ and $Per(M)$ be determinant and permanent respectively of ...
- 73
6
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141
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Is this Hankel matrix involving Bernoulli polynomials positive definite?
Let $(B_n(X))_{n \ge 0}$ denote the sequence of Bernoulli polynomials. For any couple of integers $(d,m) \in \mathbb{N}$, define the Hankel matrix
$$ H_d(m) := \left( \frac{B_{i+j+1}(m)}{i+j+1} \right)...
1
vote
1
answer
178
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Prove $ReR=R$ where $R=\prod_{i=2}^\infty \mathbb{M}_i(S_i)$ and $e$ is a certain idempotent of $R$
Let $\{S_i \}_{i=2}^\infty$ be a collection of rings. For each $i\geq 2$ let $e_i \in \mathbb{M}_i(S_i)$ be the idempotent whose (1,1)- and (2,2)-entries are $1$ and all other entries are $0$. Then $e:...
- 406
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0
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67
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Upper-bounded version of the Gale-Ryser theorem
The standard Gale-Ryser theorem is for the existence of a $(0,1)$-matrix given exact row sums $R = (r_1, \dots, r_K)$ and exact column sums $C = (c_1, \dots, c_M)$. What if we relax the column sums ...
6
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137
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Name for "continued fraction" matrices and the group they generate
Let $R$ be a ring. Is there a standard name for matrices of the form
$$
\begin{pmatrix}a & 1\\ 1 & 0\end{pmatrix}\in \mathbb{M}_2(R)?
$$
When $R=\mathbb{Z}$, these matrices arise naturally in ...
- 19.4k
2
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103
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Polynomial filter on symmetric tridiagonal matrices
During my research, I encountered a problem involving symmetric tridiagonal matrices and their eigenvectors, which I have been attempting to solve since then. So far, I have only managed to obtain a ...
- 61
3
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94
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Existence of irrational eigenvalues of a sum of representation matrices
Let $G$ be a finite non-abelian group and $H$ be a non-normal subgroup of $G$. It can be shown that there exists a non-linear irreducible unitary representation $(\rho,V)$ of $G$ such that the matrix $...
- 91
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2
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103
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Symmetric matrix inversion if diagonalization in larger dimension is known
Given an $n \times n$ orthogonal matrix $U$ (i.e., $U^T U = \mathbb I_{n\times n}$), then for an $n \times n$ diagonal matrix $D$, it is easy to verify that $ \left( U^T D U \right)^{-1} = U^T D^{-1} ...
- 343
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1
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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
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7
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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 ...
- 5,825
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1
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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
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1
answer
334
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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
0
votes
2
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186
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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
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122
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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
398
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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\...
- 942
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99
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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
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2
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118
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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
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0
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154
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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
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0
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137
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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=...
- 1,029
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131
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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
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1
answer
198
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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
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0
answers
151
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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 ...
0
votes
0
answers
84
views
Smallest eigenvalues comparison between two matrices: Seeking proof ideas
This problem stems from a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the similarity transforming matrix from ${\bf J}$ to ${\bf ...
- 75
1
vote
1
answer
414
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Sum of ranks of blocks
Let $M = \pmatrix{A & B\\ C& D}$, where $A$ is an all-one matrix. From Section 3 of Nisan & Wigderson$\color{magenta}{^\star}$,
$$\operatorname{rk} (B) + \operatorname{rk} (C) \le \...
- 551
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votes
2
answers
470
views
A prototypical problem for transfer matrix calculations in combinatorics
Let $G$ be a directed graph with self-loops on the positive integers: for every $n$, $G$ has the directed edges $(n,n)$ and $(n, n+1)$; additionally, if $n$ is even, $G$ has the directed edge $(n, n/2)...
- 9,741
5
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1
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253
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Independence of parameter for eigenvalues of periodic family of tridiagonal matrices
Consider the family of matrices $C(\ell,\theta)\in \mathbb{R}^{2\ell+2\times2\ell+2}$, where $\ell\in\mathbb{N}$ and $\theta\in\mathbb{R}$ given by
\begin{equation*}
C(\ell,\theta)=\begin{pmatrix}
...
- 153
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votes
1
answer
136
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Extending totally unimodular matrices by a column
Given a matrix $M$, we will refer to the submatrix formed by the first $k$ rows as $M([k], \cdot)$.
Let $A$ be a $m\times n$ totally unimodular matrix where $m \leq n$. We define a new $m\times (n+1)$ ...
- 123
4
votes
1
answer
161
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Fastest way to compute Cesàro limit of the powers of a stochastic matrix
Let $P$ be a (finite) stochastic matrix. Let
$$ C = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} P^k $$
be the Cesàro limit of the powers of $P$.
What is the fastest known way to compute $C$?
- 2,440
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1
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357
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Prove that a matrix is almost surely full rank
[Cross-posted from MS after 8 days without reply.] I have a real matrix $R_{ij} \in \mathbb{R}^{n \times m}$ whose entries are sampled iid from an absolutely continuous distribution $D$; a fixed real ...
- 165
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votes
1
answer
229
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Presentation of the algebraic closure of finite fields via matrices
Sorry if this question is too elementary for MO.
Let $p$ be a prime and $F_p$ a field with $p$ elements and $F_{p^n}$ the field with $p^n$ elements
Then we can choose an irreducible factor $f$ of ...
- 28.2k
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2
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277
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Factorisation of bilinear polynomial over the integers
Let $f(x, y) = axy + bx + cy + d$ be a polynomial with integer coefficients $a, b, c, d$. Is there a criterion for $f$ to be factorised as
$$ f(x, y) = (rx + s) (my + n) $$ for some integers $r, s, m, ...
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6
votes
1
answer
371
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A generalized Vandermonde matrix and spanning property
It is known that if $x_1, x_2, ..., x_n$ are all positive distinct real numbers, then the matrix
$$
\begin{pmatrix}
x_1^{a_1} & x_1^{a_2} & \cdots & x_1^{a_n} \\
x_2^{a_1} &...
- 113
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0
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145
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Can it be proved that the specific real symmetric matrix is positive definite? (Numerically confirmed)
${\bf A} \in {\Bbb R}^{n \times n}$ is a real symmetric positive definite M-matrix, whose non-diagonal entries are non-positive. Defining a composite matrix ${\bf J}\in {\Bbb R}^{2n \times 2n}$ as
$$\...
- 75
4
votes
2
answers
420
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Expansion of $(A+t B)^{-1}$ as $t\to0$
Let $A, B$ be real matrices, with $A$ symmetric, positive semi-definite, with kernel spanned by the vector full of ones, and $B$ a non-singular matrix (we do not assume that $A$ and $B$ commute).
Can ...
- 143
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votes
1
answer
485
views
Numerical tests show the smallest eigenvalue of a certain matrix remains invariant when some parameters vary – any proof ideas?
I've numerically verified that the smallest eigenvalue of a specific matrix remains invariant when some parameters change. Looking for theoretical proof approaches. Appreciate any insights.
${\bf A} \...
- 75
2
votes
2
answers
352
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Adjacency graphs and eigenvalues
I am currently reading this paper and this related paper (can also be found here), which explore the connection between Jordan normal forms and adjacency graphs.
Theorem 6 in the first paper reads
...
- 79
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136
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Constraints on building adjacency matrices
We know that all adjacency matrices are square matrices.
When our weighted directed graphs have loops or parallel edges, we obtain adjacency matrices that are, in general, asymmetric or non-Hermitian ...
- 79
0
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0
answers
59
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Triangle of integer coefficients that contain both A113340 and A113350
Let
$P(n,k)$ be A113340 (i.e., triangle $P$, read by rows, such that $P^2$ transforms column $k$ of $P$ into column $k+1$ of $P$, so that column $k$ of $P$ equals column $0$ of $P^{2k+1}$, where $P^2$...
user231922
0
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0
answers
120
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Quadratic equation involving diagonals of inverses of matrices
Let $b$ and $c$ be two real numbers, $D \in \mathbb{R}^{n \times n}$ be a diagonal matrix, $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, $Q \in \mathbb{R}^{n \times n}$ be a symmetric matrix ...
- 11
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votes
4
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571
views
How many translates of the singular‐matrix hypersurface are needed to cover $M_n(\mathbb{F}_2)$?
Let $n$ be a positive integer, and consider the hypersurface of singular $n\times n$ matrices over $\mathbb{F}_2$, denoted
$$
\mathcal{S}_n = \{X\in M_n(\mathbb{F}_2) : \det(X)=0\}.
$$
Note that
\...
- 469
0
votes
1
answer
416
views
Is this formula for a matrix block inverse in terms of the entire matrix inverse known? [closed]
I needed such a formula and when I couldn't find it on Wikipedia, I asked Claude.AI to help me derive one and this is what we came up with:
The formula: Given an invertible matrix partitioned as
$$P = ...
- 175
2
votes
0
answers
114
views
Is it possible to analyze the eigenvalue of a specific tridiagonal matrix?
I'm considering the $n \times n$ tridiagonal matrix
$$ A = \begin{pmatrix}
0 & 1 & & & \\
1 & c & 1 & & \\
...
- 21
6
votes
0
answers
113
views
Concentration bound for the number of full-rank matrices in a random subspace of matrices
Let $\mathcal{M}_{n,m}(\mathbb{F_q})$ be the set of $n$ by $m$ matrices over the finite field of order $q$, with $n \geq m$.
For a (non-trivial) subspace $V \subset \mathcal{M}_{n,m}(\mathbb{F_q})$, ...
- 271
2
votes
1
answer
175
views
Trace maximization for products of symmetric nonnegative matrices summing to $J_n$
Let $n$ be a positive integer and $A_1, A_2, \dots, A_k$ be a sequence of real symmetric $n \times n$ matrices with nonnegative entries, such that
$$A_1 + A_2 + \dots + A_k = J_n,$$
where $J_n$ ...
- 737
0
votes
0
answers
35
views
How to estimate the tail probability of a structured Frobenius norm involving Gaussian noise?
Let $ Q \in \mathbb{R}^{n \times m} $ be a known (non-random) invertible matrix, and let $ W \in \mathbb{R}^{m \times d} $ be a random matrix whose entries are i.i.d. Gaussian variables:
$
W_{kj} \sim ...
- 1
18
votes
2
answers
513
views
Number fields in fast matrix multiplication
A common approach to construct fast multiplication algorithms is to make an ansatz for the matrix multiplication tensor of fixed dimension and rank (e.g. $2 \times 2 \times 2$ and rank $7$ if we want ...
- 2,602
1
vote
0
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138
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On the reducibility of a characteristic polynomial
Given $n \times n$ Hermitian matrices $A$ and $B$, we define the polynomial,
$$g(\alpha, \beta) := \det(A + \alpha B - \beta I),$$
where $I$ is the $n \times n$ identity matrix. It may be seen as the ...
