Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,873 questions
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Finding which members of a family of (possibly infinite-dimensional) matrices have trivial null space
Background
I have a set $S$ (that is possibly infinite) and a correspondence between functions $c:S^3\to\mathbb{R}$ (I will write $c(i,j,k)$ as $c_{ijk}$) and matrices $M$ with rows indexed by $(i,j,k)...
- 509
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248
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$\mathbb R$ and $\mathbb F_2$ rank in boolean matrix product
By rank I imply rank over reals ($\mathbb R$).
I consider two $n\times n$ matrices $A,B$ having entries in $0/1$.
The product below follows usual matrix product rules except $xy$ is $AND(x,y)$ and $x+...
- 23
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194
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Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 1
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330
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Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
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94
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Large subgroups of infinite-dimensional vector spaces
Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$.
Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
- 4,547
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88
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Fast decay of eigenvector elements
Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
- 47
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69
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Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices
While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.
To ...
- 5,215
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250
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What is the computational complexity of solving a highly underdetermined system?
Let $F$ be a finite field with $q$ elements. Consider an underdetermined system of linear equations with $m$ equations and $n$ variables where $n\gg m$. What is the complexity of solving such a highly ...
- 131
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129
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Symmetry of points on unit sphere determined by relation between triples of points
Suppose we have $n$ points on the 3D unit sphere,
$X = (\pmb{r}_{1}, \pmb{r}_{2}, ..., \pmb{r}_{n})$.
I am interested in knowing to what extent the rotational symmetry of $X$ is determined by the ...
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78
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The Hilbert matrix becomes degenerate after slight modification
Let $H_{n+1}$ denote the $(n+1)\times (n+1)$ Hilbert matrix, i.e., the $(i, j)$-entry of $H_{n+1}$ is $(i+j-1)^{-1}$ with $1 \leq i, j \leq n+1$. Let $A$ denote an $(n+1)\times (n+1)$ matrix whose ...
- 1
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146
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Square root of a circulant matrix block
I'm trying to show the following:
Given the following $n\times n$ symmetric circulant matrices
$$A^*=\begin{pmatrix}
1 & -\mu_a & 0 & ...&0&-\mu_a \\
-\mu_a & 1 & -\mu_a &...
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57
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Numerically finding matrix approximation by lower-dimensional "pseudo-similar" matrix
Consider an $N\times N$ (real or complex) matrix $A$, and some $n<N$. Is there a good numerical algorithm that finds the set consisting of an $n\times n$ matrix $B$, an $n\times N$ matrix $I$, and ...
- 3,001
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299
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Question on rank of matrices over $\mathbb F_2$
$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$.
$B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$.
$T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
- 13.9k
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90
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Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
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195
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Proof negative-definiteness of a nonsymmetric and rank-deficient matrix
Consider the vectors $\mathbf{a} \in \mathbb{R}^N$ and $\mathbf{b} \in \mathbb{R}^N$ with $N>1$ and $\mathbf{a} \neq \mathbf{b}$.
The product $\mathbf{C}=\mathbf{a} \mathbf{b}^T \in \mathbb{R}^{N \...
- 1
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425
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Linear independence of vectors in Graph Theory
I have poste this question on StackExchange but there were no takers - would I be luckier on this site?
Most of this is well known, so let me just restate the corresponding Math:
Given a connected, ...
- 419
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897
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Error function of multivariate Gaussian
I'm trying to prove that a sequence of functions $(k_N)_{N\in\mathbb{N}}$
$$k_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$
where $r>0$, $k>2$ and
Edit: I have forgot to ...
- 93
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141
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Parseval's equivalent of Norm that includes a Projection matrix
I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...
- 1
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24
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Concentrate singular values by diagonal similarity
Given non-singular real matrix $A$ of size $n \times n$. I want to find a non-singular diagonal matrix $D$ such that
$$
B = D A D^{-1}
$$
has singular values $\sigma(B) = [1, \dots, 1, \lvert \det(A) ...
- 909
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42
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Relate existence of common invariant subspace of $m$ matrices to reducibility of certain polynomial
Let's recall a linear algebra fact: Let $A$ be an $n\times n$ matrix over a field $K$ and $\chi_A(t)$ be its characteristic polynomial. Then if $\chi_A(t)$ is reducible, $A$ would have a proper ...
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99
views
Unimodular matrices fixing $(1, 1, \cdots, 1)$
What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
- 356
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67
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The minimum number of polynomial equations the components of linearly dependent vectors must satisfy
Context:
Consider $m<n$ vectors $v_1,\dotsc,v_m\in\mathbb{C}^n$ with complex components. We can study if they are linearly dependent by constructing the following matrices. First the $n\times m$ ...
- 344
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1
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533
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Follow up: Show that these vectors are linearly independent almost surely
I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I want to discuss regarding it. Unfortunately I can'...
- 344
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149
views
L_q matrix inequality
The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
- 101
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41
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Iterative algorithm for obtaining similarity
Let $x_1,x_2,\ldots,x_M$ be $M$ non-negative variables. Moreover, assume that $f_m(x_m)=\frac{x_m}{1+\sum_{n}\beta_{n}^{(m)}x_n}$ be $M$ fractional functions with non-negative constants $\beta_{n}^{(m)...
- 287
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202
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Ratio of maximum to minimum value
Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then ...
- 61
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113
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Error bounds on the expansion of square root of matrix
I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
- 427
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92
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Linear independence of Wishart matrices
Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
- 123
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190
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Does the functional square root of the cosine admit a vector-based interpretation?
In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$
The functional square root of the cosine has at ...
- 4,829
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108
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Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
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127
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Maximizing a vector after a series of matrix multiplications
Problem Statement
Let's say we have a set of $n\times n$ matrices $X=\{M_1,\ldots,M_r\}$ and weights of these matrices $\{w_1,\ldots,w_r\}$ along with a set of "initial vectors" $\{v_1,\...
- 509
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70
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Birkhoff's theorem for hypergraphs
Birkhoff's theorem says that, in a bipartite graph $G$ in which both sides have size $n$, any fractional matching of size $n$ can be presented as a convex combination of integral matchings of size $n$ ...
- 3,549
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61
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Combining quadratic and linear matrix terms into a quadratic term
Given
$$ C = AFA^T + A\bar{F} $$
where $A = [A_1 A_2]$, $F = \begin{bmatrix}
F_1 & F_2 \\
F_2^T & F_3
\end{bmatrix}$, $\bar{F} = 2 \begin{bmatrix}
\bar{F}_1 \\
\bar{F}_2
\end{bmatrix}$ such ...
- 1
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45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
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236
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Eigenvectors of a matrix
Let $M$ be a square matrix of order $n\times d$. Let $\xi_{1},\dots,\xi_{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have
$$\xi_{i}=(\lambda_1, 0,...
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Is the free algebra over an operad an algebra over that operad?
I'm asking here this question I asked on MSE that got no answers.
Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...
- 499
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Is there a method to find a vector that optimizes a Rayleigh quotient over a subspace?
Let $M\in\mathbb{C}^{n\times n}$ be an arbitary Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
Is there a method to find vectors $y,z\in E$ such that
$$\dfrac{y^*My}{y^*y}=\sup_{x\in E\\...
- 596
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141
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Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
- 356
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86
views
Eigenvalue inequality
Let
$g(\boldsymbol{\theta},\boldsymbol{\theta_0}) = trace [
\boldsymbol{\Omega{(\boldsymbol{\theta})}}^{-1} \boldsymbol{\Omega{(\boldsymbol{\theta_0})}}]-ln[det(\boldsymbol{\Omega{(\boldsymbol{\theta}...
- 31
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149
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Upper-triangular matrices as union of centralizers of cyclic elements
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The ...
- 463
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30
views
Signs of difference matrices (sum of submatrices)
Given matrix $A \in \mathbb{R}^{m \times n}$, are there any results related to its difference array
$$A^* \triangleq \left[sign(a_{i,j} + a_{r, s} - a_{r, j} - a_{i, s})\right]_{i<r, j<s}?$$
Or ...
- 75
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45
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On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
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91
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Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
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159
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How to solve a non-local self-consistent equation
I have been struggling lately with solving numerically an equation of the form:
$$ g(x\pm x_{0}) = F[ g(x) ] $$
where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...
- 123
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47
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"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
- 2,712
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101
views
Find occurrences of certain matrix inside a matrix
This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching.
We know for ...
- 101
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132
views
How can GL(n) acts on the determinant polynomial?
I'm reading Landsberg's paper, which provide an introduction to geometric complexity theory. At chapter 2 of this paper, the author defined the following objects:
Let $W = \mathbb {C}^{n^2}$, $det_n \...
- 47
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0
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50
views
Generalized eigenvectors product
Let's consider a real square matrix $A$ with eigenvalues $\lambda_n$ and eigenvectors $\mathbb x_n$, i.e. $A \mathbb x_n = \lambda_n \mathbb x_n$.
Suppose there are some generalized eigenvectors $\...
- 141
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0
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50
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restriction of a formula with matrix inverse multiplied by a vector
I'm trying to reproduce a proof from this paper but I'm stuck in one point (Lemma 6). The general subject is bayesian model for multi-armed bandit problem solved with Thompson sampling.
I think I ...
- 1
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0
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33
views
Find a complex matrix on a unit sub-spheres
I am new to optimization theory. I have a following question. For a given $X = [x_1 x_2 \ldots x_N] \in \mathbb{C}^{N \times N}$, where $x_i \in \mathbb{C}^{N\times 1}$ for $i \in \{1,\ldots,N\}$, $U =...
