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.
3,106
questions
2
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A line bundle over the manifold of singular matrices
According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$
So we ...
1
vote
1
answer
364
views
A geometric property of singular matrices
Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let $M=...
4
votes
1
answer
214
views
Spectrum of primitive nonnegative integer matrices
Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$.
Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
2
votes
2
answers
605
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Matrices with real spectrum
Assume you have a non-symmetric real square matrix all of whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix?
EDIT: Is it at least similar to ...
1
vote
0
answers
260
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Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
0
votes
0
answers
123
views
A quantity associated with an algebraic variete
Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that ...
1
vote
1
answer
348
views
Matrix Submodular Inequality
Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property ...
19
votes
2
answers
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Why is a matrix pencil called a pencil?
I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on.
I am aware that even Gantmacher 1959 has this terminology however I don't know ...
6
votes
1
answer
790
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Jordan decomposition of the tensor product of two matrices
I asked this question on Math.SE here, but did not get a lot of attention.
I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
10
votes
1
answer
271
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Is the center of the automorphism group of a von Neumann algebra M trivial whenever M is a factor?
Question: Is the center of the automorphism group of a von Neumann algebra $\mathscr{M}$ trivial (=$\{\mathrm{id}\}$) whenever $\mathscr{M}$ is a factor (=$\mathscr{M}$ has center $\{\lambda I; \...
3
votes
1
answer
274
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Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements
Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...
8
votes
0
answers
5k
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Partitioned inverse 3x3 block matrix
We know that matrices can be inverted blockwise by using the following analytic inversion formula:
\begin{equation}
\begin{bmatrix} \mathbf{A} & \mathbf{C^T} \\ \mathbf{C} & \mathbf{D} \end{...
1
vote
0
answers
393
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Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
1
vote
0
answers
108
views
An exact fraction of a matrix
Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that $\...
3
votes
2
answers
279
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Rank changes with matrix edits
Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...
0
votes
0
answers
80
views
Bits of precision matrix reconstruction
We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
0
votes
0
answers
159
views
Way to parameterise sparse multi diagonal matrix
I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$
where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is
$$
\Lambda = \begin{bmatrix}
x &...
9
votes
2
answers
681
views
Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$The exceptional isomorphism $\Spin(5,1)\simeq \SL(2,\mathbb{H})$ is well-known, and I can find references that ...
4
votes
0
answers
89
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Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?
Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries).
Can we partition this union into at most $n$ rectangles?
I think it's pretty ...
15
votes
2
answers
8k
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Is there a standard notation for off-diagonal transpose?
Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$,
its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$.
But is there a conventional way of notating the matrix
...
6
votes
1
answer
653
views
Some calculus in the orthogonal group $O(n)$
How can one compute each of the following matrices, explicitly:
$$\int_{O(n)} e^{g}dg$$ or
$$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$
What is the explicite entries of the resulting ...
1
vote
1
answer
321
views
A (possible) equivalent relation on the space of vector bundles
Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and $...
4
votes
2
answers
259
views
An algorithm to compare two representations of a simple Lie algebra?
I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...
5
votes
2
answers
516
views
Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$
I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$)....
0
votes
0
answers
247
views
Zariski dense subgroups and conjugates
Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
6
votes
1
answer
617
views
Subadditivity of the square root for matrices
For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?
If not, ...
1
vote
1
answer
145
views
Are certain normal matrices circulant? (Part 2)
Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $
A^TA=\begin{pmatrix}
a & b \\
b & \ddots
\end{pmatrix}$, for $b>0$.
As a user observed in the solution of Part 1 ...
1
vote
3
answers
1k
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Simple Spectrum of Jacobi matrices
I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...
5
votes
0
answers
317
views
Eigenvalues of Random Regular Bipartite Graphs
I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
0
votes
1
answer
202
views
Are these particular kinds of matrices well known?
Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are $\pm ...
4
votes
5
answers
3k
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About adding a negative definite rank-1 matrix to a symmetric matrix
If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...
1
vote
0
answers
107
views
Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
6
votes
2
answers
437
views
Is a normal matrix satisfying $A^TA=...$ circulant?
Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that
$$
A^TA=\begin{pmatrix}
a & b & \cdots & b\\
b & a & \ddots & \vdots\\
\...
1
vote
1
answer
115
views
QR decomposition of matrix [closed]
I have matrix $M = \begin{pmatrix} A & B \\ B^T & 0\end{pmatrix}$, where $A$ is $N\times N$, $B$ is $N\times 2$ and I have $Q$, $R$ such that $A = QR$. What is the fastest way to find $Q'$ and ...
5
votes
0
answers
254
views
Existence of a matrix product from its eigenvalues
Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia):
$$
\lambda^\downarrow(A) \cdot \lambda^\...
2
votes
1
answer
206
views
Are there good ways of relating a minor to the full determinant?
Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
-1
votes
1
answer
1k
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How to show the square root function of a positive semidefinite matrix is differentiable? [closed]
How to show the square root function of a positive semidefinite matrix is differentiable?
In this context PSD means symmetric PSD.
4
votes
0
answers
329
views
Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?
Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.)
I want to calculate or bound $\det (...
2
votes
0
answers
661
views
Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix
I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...
8
votes
3
answers
2k
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Transforming a binary matrix into triangular form using permutation matrices
I am interested in the complexity of the following problem:
Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix?
I am also interested in ...
1
vote
1
answer
1k
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The norm of a Finite Hilbert matrix
Let $H$ be an $n\times n$ Hilbert matrix,
$$h_{ij}=(i+j-1)^{-1}.$$
The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left \|...
12
votes
3
answers
3k
views
How do we show this matrix has full rank?
I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988):
Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
j+...
5
votes
1
answer
497
views
Finding sparsest solution of a linear system
I want to find the solution with most zero-components for the following problem:
$Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...
1
vote
1
answer
344
views
Maximizing a certain concave function over a non-convex set
I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ i=1,\cdots,...
11
votes
3
answers
8k
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Eigenvectors of a symmetric positive definite Toeplitz matrix
I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better.
Although I assumed this would be a well ...
6
votes
1
answer
272
views
Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$
Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
9
votes
0
answers
210
views
Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?
--Updated description--
I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test.
Suppose there is a vector space $W=\operatorname{span}...
16
votes
2
answers
730
views
ULU Decomposition of a matrix
Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...
0
votes
0
answers
213
views
Canonical forms of symmetric/skewsymmetric quaternionic matrix
$A$ belongs to $n$-dimensional quaternion symmetric matrix, in the sense that $A=A^T$, where $T$ means transpose. Under transformation $U$, $A\rightarrow U\cdot A\cdot U^T$, where $U$ is $n$-dim ...
1
vote
0
answers
136
views
when can I say that $UV^T$ is a permutation matrix? [closed]
suppose we have two p.s.d matrices A and B: so we can diagonalize them like this:
A= $UΛU^T$ and $B=VΣV^T$
1: on what condition for $A$ and $B$ I can say that $UV^T$ is a permutation matrix?
2: how ...