Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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Approximating symmetric matrices by symmetrized low rank matrices

Fix an integer $k$, and suppose $M$ is a real symmetric $n\times n$ matrix admitting a decomposition: $$ M = A + A^t + B $$ with $\mathrm{rank}(A)=k$ and: $$ \|B\|_2 \ll \lambda_{1}(M_{|\mathrm{range}(...
alesia's user avatar
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1 answer
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On rank one torsion-free modules over local rings [closed]

Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes_A k \le 1$? If not, is there a known upper-bound ...
Ron's user avatar
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2 votes
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Isomorphic quiver representations "after adding some zeros"

Let $Q$ be a quiver, with dimension vector $d$ and let $e$ be another dimension vector, such that $d_v\leq e_v$ for every vertex $v$ of $Q$. If $M$ is a $K$-representation of $Q$ of dimension vector $...
WangWei's user avatar
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8 votes
3 answers
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Optimization problem with determinant as objective

Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem \begin{align}...
dineshdileep's user avatar
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2 votes
0 answers
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Distinct singular values of a matrix perturbed with a symmetric matrix

Is it always possible to perturb a matrix using a symmetric matrix such that its singular values become distinct? Formally: Let $A$ be an arbitrary (finite dimensional) complex square matrix. Let $\...
Dominique Unruh's user avatar
5 votes
1 answer
1k views

In a large sparse matrix, how many eigenvalues/eigenvectors are “spurious”?

In a large (possibly above $5000\times 5000$) matrix, the problem of finding all the eigenvalues and eigenvectors can be solved using iterative methods (Arnoldi, Lanczos etc.). However, there seems to ...
user avatar
12 votes
2 answers
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Eigenvalue perturbation theory via Feynman diagrams

Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
thedude's user avatar
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A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular

A paper I'm reading in representation theory states the following result: Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
D_S's user avatar
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4 votes
1 answer
408 views

Least square solution to $AXB+CXD=E$

I am trying to find the least-squares solution $X$ of the following matrix equation $$AXB+CXD=E$$ Of course, I know that this equation can be written in the form $$(B^T \otimes A+D^T \otimes C) \...
dave2d's user avatar
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5 votes
4 answers
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Why do some linear cellular automata over $Z_{2}$ on the torus have small order?

At https://dmishin.github.io/js-revca/index.html, you can play around with reversible cellular automata. I noticed that on that site, that for the reversible linear cellular automata (which I have ...
Joseph Van Name's user avatar
14 votes
1 answer
432 views

Similar matrices over $\mathbb Z_p$

Let $A$ and $B$ be two $n \times n$ matrices with entries in $\mathbb Z_p$, the $p$-adic integers. Is it true that $A$ and $B$ are conjugate iff they're conjugate over $\mathbb Q_p$ and over $\mathbb ...
Nick Addington's user avatar
5 votes
1 answer
888 views

The spectrum of the discrete Laplacian

Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$ On this we ...
Dr. House's user avatar
5 votes
1 answer
218 views

Equivalence of $G$-invariant symplectic forms

Let $V$ be a finite-dimensional complex vector space with a linear action of a complex reductive group $G$. Suppose that $\omega_0$ and $\omega_1$ are two $G$-invariant complex symplectic bilinear ...
user115839's user avatar
8 votes
2 answers
731 views

Matroids of rank two

I am interested in matroids of rank two and would like to understand how interesting/big this class of matroids is. I know that the 2-uniform matroid on (k+2) elements is not representable over any ...
Quentin Fortier's user avatar
5 votes
1 answer
272 views

Natural explanation for a matrix identity

I recently came across this curious fact in some calculations with the strain tensor in fluid mechanics: Let $A$ be an antisymmetric 3 by 3 matrix and $S$ be a traceless symmetric 3 by 3 matrix. Then ...
Fan Zheng's user avatar
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7 votes
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Counting 0-1 $n\times n$ matrices with a given rank r

What is the number $N$ of $n \times n$ $0$-$1$ matrices with rank $k$? I read this sequence is "OEIS A064230 Triangle $T(n,k)$ = number of rational (0,1) matrices of rank $k$ ($n\ge 0$, $0\le k\le ...
Penelope Benenati's user avatar
1 vote
1 answer
228 views

Perron-frobenius theorem for Hermitian matrices

As I'm working on a concept in linear algebra, I want to know is there a perron-frobenius theorem for Hermitian matrices? I tried to find something on the web but I couldn't find anything. Bests.
A. Mpi's user avatar
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6 votes
0 answers
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formalization of coordinate-free linear algebra in a proof assistant

I am aware of projects that formalize linear algebra in existing proof assistants (i.e. Coq), but it seems like most of them are based on matrices. I was wondering if it's done in a coordinate-free ...
D. Huang's user avatar
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-1 votes
1 answer
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A simple matrix multiplication query [closed]

The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
Turbo's user avatar
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1 vote
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Difference between largest two eigenvalues of a graph Laplacian

The difference between the smallest eigenvalue and the next-smallest of a graph Laplacian (equivalently, the difference between the largest and next-largest of the random walk Markov chain on the ...
Ben Golub's user avatar
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4 votes
0 answers
923 views

Generate non-negative linear combinations of non-negative vectors with different supports

(I will not be surprised if this problem has been solved and/or has a trivial solution – I just do not know the right terminology to google for it.) So the problem is as follows. I have an $m \...
Yauhen Yakimenka's user avatar
6 votes
1 answer
1k views

Computing kernels of maps of modules over a finitely presented algebra

I have the following problem: I have an associative (noncommutative) algebra $A$ defined over a rational function field $k = \mathbb{Q}(\delta, \lambda)$. $A$ is given by a presentation in terms of ...
Calvin McPhail-Snyder's user avatar
7 votes
1 answer
293 views

Argument principle for matrices

Let $f,g$ be entire functions, then the argument principle teaches us that $$\frac{1}{2\pi i}\int_{\mathbb{C}} g(z) \frac{f'(z)}{f(z)} dz$$ is equal to $g$ evaluated at the zeros of $f.$ Now, let ...
Zehner's user avatar
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0 answers
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Changing Couplings of Discrete Random Variables

Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
The Substitute's user avatar
5 votes
0 answers
78 views

Generalized rational form

Fix a field $k$. It is well known that any square matrix is similar to a block-diagonal matrix such that every block is of the following form: $1$ in the subdiagonal, a bunch of numbers in the last ...
Bissaque's user avatar
11 votes
0 answers
303 views

Jaffard's theorem - finite matrices

For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+\left|k-l\right|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,...
Ozzy's user avatar
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5 votes
1 answer
651 views

Is every real matrix conjugate to a semi antisymmetric matrix?

Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
Ali Taghavi's user avatar
1 vote
0 answers
113 views

smallest singular value over invertible sub-matrices

Consider the matrix $M = \begin{bmatrix} A & A B \end{bmatrix} \in R^{n \times (n+m)}$, with $A \in R^{n\times n}$, $B \in R^{n \times m}$, $m < n$, $m > 1$, $A$ symmetric positive definite. ...
yon's user avatar
  • 303
4 votes
1 answer
304 views

How can we find a monic polynomial with the smallest degree in left ideal of $\mathrm{Mat}(F[x])$?

Let $F$ be a finite field, $R=F[x]$ be a polynomial ring and $K = \mathrm{Mat}_n(R)$ be a full matrix ring over $R$. We identify the ring $K$ with the ring $\mathrm{Mat}_n(F)[x]$, for example $$ \left(...
Mikhail Goltvanitsa's user avatar
2 votes
0 answers
265 views

Maximum spectral norm of matrices with given anti-Hermitian part and Hermitian part's spectrum

Let $M\in M_n(\mathbb C)$ be a $n\times n$ matrix over the complex field. It can be written uniquely as $M=H+A$, where $H=H^*$ denotes its Hermitian part and $A=-A^*$ its anti-Hermitian part. Its ...
francesco999's user avatar
3 votes
0 answers
189 views

Is there a reasonable way to check intersection of these set of vectors?

Given $a,m,n,t\in\Bbb Z$, with $n=m^t$ and $a$ arbitrary, and given $\mathbb{Z}$-linearly independent vectors $v_1,\dots,v_n\in\Bbb Z^n$, and an arbitrary vector $w\in\Bbb Z^n$, such that $$\langle ...
Turbo's user avatar
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3 votes
1 answer
134 views

Non alternative $k$-linear maps vanishing on $\sum x_i=0$

Assume that $V$ is a finite dimensional real vector space of dimension $n$. Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple ...
Ali Taghavi's user avatar
0 votes
1 answer
509 views

Directed graph cycles and the inverse of a weighted adjacency matrix

Let us view a matrix $B \in \mathbb{R}^{n \times n}$ as the weighted adjacency matrix of a directed graph $G$, i.e. there is an edge $i \to j$ in $G$ if $B_{ij} \neq 0$. Assume further that $B$ does ...
passerby51's user avatar
  • 1,639
2 votes
0 answers
113 views

Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$

Suppose we have a matrix $$T\in\Bbb K^{n^k\times m}$$ and a target vector $v\in\Bbb F_q^m$ where $m<n^k$ and $1<k$ holds. We need to find $k$ vectors $u_1,\dots,u_k\in\Bbb K^n$ such that $$v=...
Turbo's user avatar
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3 votes
2 answers
1k views

Roots of quadratic vector equation

Given $$A_{i j k}X_j X_k + B_{ij} X_j + C_i = 0$$ where $A_{ijk}$, $B_{ij}$, and $C_i$ are arbitrary real numbers for all $i$, $j$, $k$ which are $N$-dimensional indices, such that $A_{ijk}=A_{ikj}$ ...
Bence Kocsis's user avatar
3 votes
1 answer
81 views

Polytopes that are just defined by ordering the variables

I am working with a polytope with a very specific structure, namely that it is characterized entirely by placing the variables, or the variables plus constants, or just constants, in a particular ...
Tom Solberg's user avatar
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0 votes
0 answers
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Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]

Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation $(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
Pierre's user avatar
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1 vote
1 answer
211 views

Matrix golf puzzle: enumerate a series by matrix multiplication

It is super easy to find matrices $X_0$, $F$ and $H$ such that $H F^n X_0$ is equal to $n$-nth element of the sequence $0,1,0,1,0,1,0,1,0,1,0,1,...$ Maybe it is a slightly harder challenge to find ...
O.Rerla's user avatar
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1 vote
1 answer
715 views

Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets

Assume $x$ is a variable belongs to $\mathbb R \setminus \{ 0,-1,+1 \}$ and consider for all $i, j \in \mathbb N$, $$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$ then for all $n \in \mathbb N$ the ...
Ahmad Jamil Ahmad Masad's user avatar
4 votes
2 answers
832 views

Steady state Kalman filter

My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from. Kalman filter allows us to estimate state at time $t$ as ...
O.Rerla's user avatar
  • 67
7 votes
2 answers
411 views

Purely complex eigenvalue of matrix product

Here is a question which arises from physics. Let $A$, $B$ be two symmetric real-valued matrices. What conditions should the matrices meet to make $AB$ has a pure complex eigenvalue ($Im(\lambda) \...
Max Borovkov's user avatar
1 vote
1 answer
189 views

Linear equation for a great circle on a (multidimensional) sphere

Can we introduce independent coordinates on a sphere such that any great circle could be represented as a linear equation (like line on the plane)? If yes, what is a generalization for higher ...
makkostya's user avatar
  • 415
2 votes
0 answers
2k views

Sufficient conditions for positive semidefiniteness of block matrix

$\newcommand{\Re}{\mathbb{R}}$I m looking for sufficient conditions that may guarantee positive semidefiniteness (PSD) of a block matrix $$A = \begin{bmatrix} A_{1,1} & \cdots & A_{1,n} \\ \...
jesusbriales's user avatar
3 votes
0 answers
76 views

A concentration problem of product of matrices

Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and ...
ie86's user avatar
  • 195
1 vote
0 answers
53 views

Maximum number of matrices satisfying given rank conditions

Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that (i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$; (ii)...
SGC's user avatar
  • 147
10 votes
2 answers
2k views

Is there a standard name for (non-square) matrices with orthonormal columns?

One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$). Is there a name ...
Federico Poloni's user avatar
13 votes
1 answer
917 views

Axiom(s) of choice and bases of vector spaces

I'm not sure this question is more suitable for MO or for MSE, so feel free to move it to MSE if necessary. I work here in ZF theory. Consider the following statements: $(C)$ Axiom of choice: for ...
GreginGre's user avatar
  • 1,661
3 votes
1 answer
388 views

What's the best orthonormal matrix to align two matrices in the operator norm sense?

Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that \begin{equation} UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F}, \end{equation} where $USV^\top$ is ...
Wuchen's user avatar
  • 505
2 votes
2 answers
432 views

Entrywise modulus matrix and the largest eigenvector

Disclaimer. This is a cross-post from math.SE where I asked a variant of this question two days ago which has been positively received but not has not received any answers. Let $A$ be a complex ...
Julian's user avatar
  • 613
1 vote
2 answers
470 views

Closed form for integral of function of a symmetric positive definite matrix

Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm. Is it possible to evaluate the following integral in closed form? ...
Abhishek Halder's user avatar

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