Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,664
questions
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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:
$$
\|B\|_2 \ll \lambda_{1}(M_{|\mathrm{range}(...
-1
votes
1
answer
115
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On rank one torsion-free modules over local rings [closed]
Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes_A k \le 1$? If not, is there a known upper-bound ...
2
votes
0
answers
70
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Isomorphic quiver representations "after adding some zeros"
Let $Q$ be a quiver, with dimension vector $d$ and let $e$ be another dimension vector, such that $d_v\leq e_v$ for every vertex $v$ of $Q$. If $M$ is a $K$-representation of $Q$ of dimension vector $...
8
votes
3
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2k
views
Optimization problem with determinant as objective
Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem
\begin{align}...
2
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0
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170
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Distinct singular values of a matrix perturbed with a symmetric matrix
Is it always possible to perturb a matrix using a symmetric matrix such that its singular values become distinct?
Formally:
Let $A$ be an arbitrary (finite dimensional) complex square matrix. Let $\...
5
votes
1
answer
1k
views
In a large sparse matrix, how many eigenvalues/eigenvectors are “spurious”?
In a large (possibly above $5000\times 5000$) matrix, the problem of finding all the eigenvalues and eigenvectors can be solved using iterative methods (Arnoldi, Lanczos etc.). However, there seems to ...
12
votes
2
answers
1k
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Eigenvalue perturbation theory via Feynman diagrams
Suppose I have a matrix given by a sum
$$A=D+\epsilon B$$
where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
0
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0
answers
228
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A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
4
votes
1
answer
408
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Least square solution to $AXB+CXD=E$
I am trying to find the least-squares solution $X$ of the following matrix equation
$$AXB+CXD=E$$
Of course, I know that this equation can be written in the form
$$(B^T \otimes A+D^T \otimes C) \...
5
votes
4
answers
573
views
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 ...
14
votes
1
answer
432
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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 ...
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 ...
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 ...
8
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2
answers
731
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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 ...
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 ...
7
votes
0
answers
276
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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 ...
1
vote
1
answer
228
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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.
6
votes
0
answers
291
views
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 ...
-1
votes
1
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303
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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 ...
1
vote
0
answers
394
views
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 ...
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 \...
6
votes
1
answer
1k
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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 ...
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 ...
0
votes
0
answers
92
views
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 $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
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 ...
11
votes
0
answers
303
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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
$$
A^{-1}(k,...
5
votes
1
answer
651
views
Is every real matrix conjugate to a semi antisymmetric matrix?
Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
1
vote
0
answers
113
views
smallest singular value over invertible sub-matrices
Consider the matrix $M = \begin{bmatrix} A & A B \end{bmatrix} \in R^{n \times (n+m)}$, with $A \in R^{n\times n}$, $B \in R^{n \times m}$, $m < n$, $m > 1$, $A$ symmetric positive definite.
...
4
votes
1
answer
304
views
How can we find a monic polynomial with the smallest degree in left ideal of $\mathrm{Mat}(F[x])$?
Let $F$ be a finite field, $R=F[x]$ be a polynomial ring and $K = \mathrm{Mat}_n(R)$ be a full matrix ring over $R$. We identify the ring $K$ with the ring $\mathrm{Mat}_n(F)[x]$, for example
$$
\left(...
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 ...
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 $$\langle ...
3
votes
1
answer
134
views
Non alternative $k$-linear maps vanishing on $\sum x_i=0$
Assume that $V$ is a finite dimensional real vector space of dimension $n$.
Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple ...
0
votes
1
answer
509
views
Directed graph cycles and the inverse of a weighted adjacency matrix
Let us view a matrix $B \in \mathbb{R}^{n \times n}$ as the weighted adjacency matrix of a directed graph $G$, i.e. there is an edge $i \to j$ in $G$ if $B_{ij} \neq 0$. Assume further that
$B$ does ...
2
votes
0
answers
113
views
Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$
Suppose we have a matrix $$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=...
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 $A_{ijk}=A_{ikj}$ ...
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 ...
0
votes
0
answers
100
views
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 = ...
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 ...
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 ...
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 ...
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) \...
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 ...
2
votes
0
answers
2k
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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} \\ \...
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 ...
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)...
10
votes
2
answers
2k
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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 ...
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 ...
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 ...
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 ...
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?
...