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
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On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
0
votes
0
answers
50
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Possible covariance matrices of predictions of a stationary process
Let $X_t$ be a discrete time zero-mean real-valued stationary Gaussian process adapted to a $\sigma$-field ${F}_t$. Let us define
$Z_{t,j} \equiv \mathbb{E}[X_{t+j}|{F}_t]$
I am interested in ...
1
vote
1
answer
4k
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Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks
Consider the $(m+n) \times (m+n)$ block matrix
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$
I need references where they are talking about the relation between the eigenvalues of $M$ ...
8
votes
2
answers
506
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Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
0
votes
0
answers
86
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Expectation of the operator norm of projection of a random permutation matrix
Assuming I have a fixed dimension $p$ subspace of $\mathbb{R}^d$ orthogonal to $1^d$ and $VV^\top$ with $V \in \mathbb{R}^{d \times p}$ is the orthogonal projection to the subspace.
What bound can I ...
5
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0
answers
153
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Is the matrix multiplication exponent $\omega$ independent from the choice of the base field
The matrix multiplication exponent, usually denoted by $\omega_{F}$, is the smallest real number for which any two $n\times n$ matrices over a field $F$ can be multiplied together using ${\...
0
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0
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63
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Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant
Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse.
...
7
votes
1
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746
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Remarkable recursions for the A204262
Let $a(n)$ be A204262 i.e. permanent of the matrix $n\times n$ with elements $\min(i,j)$.
Let
$$
f_{n,\ell}(x)=g_{n,\ell}(x)+f_{n,\ell-1}(\ell)-g_{n,\ell}(\ell), \\
g_{n,\ell}(x)=\int (n-\ell)^2 f_{n-...
3
votes
1
answer
153
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Elastic network model Hessian rigid body motion 0 eigenvalues
Based on a method that apparently seems to be widely used in computational chemistry (cf https://en.wikipedia.org/wiki/Anisotropic_Network_Model)
Trying to build a very simple model with 3 atoms ...
1
vote
1
answer
72
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One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
0
votes
0
answers
44
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A particular selection of rows in upper triangular matrices
Let $A$ be a strictly upper triangular $n\times n$ matrix whose entries are either 0 or 1 (diagonal entries are all 0) with the nullity $m<n$.
Let us denote $R_j$ and $C_j$ with the rows and ...
3
votes
1
answer
127
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On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
1
vote
0
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106
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Hard instances for this graph isomorphism algorithm based on powers of weighted adjacency matrices?
In short, I found an algorithm for GI and the only hard instances
I found so far are non-isomorphic strongly regular graphs with
large automorphism groups.
Q1 What are hard instances for the ...
0
votes
0
answers
64
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Approximate solution problem of rank-one modification matrix secular equation
In Golub's paper , page 327,the eigenvalues of a rank-one modification of a $n\times n$ symmetric matrix can be computed by findng the zeros of the secular equation
\begin{equation*}
w(\lambda_j)=...
5
votes
1
answer
104
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Estimating a symmetric positive-definite matrix from list of matrix vector products
I have a symmetric positive definite matrix (hessian) $H$ which is unknown and expensive to compute explicitly (circa 30*30)
Indirectly in my code I have a growing list of pairs of unit vectors $u_i$, ...
6
votes
1
answer
255
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The combinatorics of the Nullstellensatz for the variety of nilpotent matrices
Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
8
votes
6
answers
7k
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How to approximate a solution to a matrix equation? [closed]
Suppose a matrix equation $Ax = b$ has no solution ($b$ is not in the column space of $A$)
How can I find a vector $x^\prime$ so that $Ax^\prime$ is the closest possible vector to $b$?
1
vote
0
answers
50
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Closed form of the product of these $2\times 2$ matrices
I have a series of $2\times 2$ matrices denoted by
$$
M_j=\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}+a_j\begin{pmatrix}-e^{i\theta} & e^{i\theta} \\ e^{-i\theta} & ...
2
votes
2
answers
191
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Questions regarding answer to complex symmetric square root of a complex symmetric invertible matrix
I am looking at showing that a complex symmetric invertible matrix always has a complex symmetric square root and I refer to this Q&A for the answer to this question. I am little confused at the ...
1
vote
2
answers
284
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Eigenvectors of a non-symmetric rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $...
2
votes
0
answers
129
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Permutation similarity of matrices with many distinct entries
This is related to graph isomorphism.
Here matrices are square $n \times n$ with non-negative integer entries.
Two matrices $A,B$ are permutation similar if there exist
permutation matrix $P$ such ...
5
votes
2
answers
434
views
Spectral radius of a rank-1 perturbation
Suppose that $\bf A$ is an $n \times n$ matrix, and $\bf u$ and $\bf v$ are vectors. The matrix determinant lemma lets us easily compute the determinant of ${\bf A} + {\bf u} {\bf v}^\top$, while the ...
3
votes
0
answers
137
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Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
2
votes
1
answer
177
views
Matrix function as gradient
Let $S_n^{++}(\mathbb{R})$ be the space of $n \times n$ symmetric positive definite matrices. For $M \in S_n^{++}(\mathbb{R})$ consider the function $f: X \in S_n^{++}(\mathbb{R}) \mapsto M X^{-1} M$.
...
0
votes
0
answers
45
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"Canonically" rotate a path/trajectory/sequence of points
I have a sequence of $n$ points in $\mathbb{R}^3$: $$P_0, P_1, P_2, \ldots, P_n$$ where $ P_i = (x_i, y_i, z_i).$ We can assume, that they are "centered", i.e. the mass center (average) is ...
5
votes
2
answers
283
views
Maximal eigenvalue of a correlation matrix with some entries fixed as zeros
Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
4
votes
0
answers
160
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Dimensionality reduction preserving cyclic traces
Suppose that I have $n$ matrices $A_1, \ldots, A_n \in \mathbb{R}^{m \times m}$ with $m \gg n$. Can I find $n$ new matrices $B_1, \ldots, B_n \in \mathbb{R}^{n \times n}$ that have the same 3-way ...
4
votes
1
answer
449
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An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
2
votes
1
answer
159
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For a divergence free smooth vector field $v : \mathbb{R}^3 \to \mathbb{R}^3$, how to find the commutator form of the matrix $A=(\partial_i v_j)$?
The question is as in the title.
I know that a traceless matrix can be written as a commutator of two matrices.
Then, let $v : \mathbb{R}^3 \to \mathbb{R}^3$ be a divergence-free smooth vector field. ...
3
votes
0
answers
81
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How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian.
Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting
$$f_{...
15
votes
7
answers
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Binary matrices with constant row and column sums
My question is about $m \times n$ binary matrices (aka $\{0,1\}$-matrices), whose rows all sum to the same value, and whose columns all sum to the same value (but these two values may be different).
...
2
votes
1
answer
124
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Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
17
votes
2
answers
1k
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The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
3
votes
1
answer
162
views
Positive-definite block matrix with constant block sums
Given two natural numbers $n$ and $m$, suppose that $A$ is an $nm \times nm$ real nonnegative matrix. Seeing $A$ as a block matrix where each block has size $m\times m$, suppose that the sum of the ...
1
vote
0
answers
40
views
About nilpotent Jordan algebras, matrix representations and formally real algebras
Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space.
Notice first that an associative ...
4
votes
2
answers
2k
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Determinant of structurally symmetric $n$-banded matrix?
Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:
$$ \left[\begin{matrix}
c_{0} & s_{0} & 0 &...
1
vote
1
answer
99
views
Orthonormal matrices with columns that switch signs
Consider an orthonormal matrix $W\in\mathbb{R}^{2n\times 2n}$ that satisfies the "abs property" $$|w_i|^T |w_{i+n}|=1$$ for all $i \in \{1,2,\ldots,n\}$, where $w_i \in \mathbb{R}^{2n}$ is ...
2
votes
1
answer
151
views
Is there an abstract theory of multi-spectral radii?
There seems to be many valid ways of generalizing the notion of the spectral radius $\rho(A)$ of a complex matrix $A$ to spectral radii of multiple operators. I am wondering if there is an abstract ...
8
votes
3
answers
7k
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Spectrum of an adjacency matrix
The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...
35
votes
0
answers
905
views
Orthogonal vectors with entries from $\{-1,0,1\}$
Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $...
5
votes
1
answer
292
views
Diagonalization of symmetric matrices of functions
I asked this question some time ago in MSE but I didn't recieved any feedback.
https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions
This problem arised ...
3
votes
1
answer
207
views
Existence of a matrix with bounded entries and large smallest singular value
Is the following statement true?
For all $n$ large enough, there exists an $M_n \in [-1,1]^{n \times n}$ such that the smallest singular value of $M_n$, $\sigma_n(M_n) \gtrsim \sqrt{n}$.
If $n$ is ...
0
votes
2
answers
566
views
Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation
Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
6
votes
1
answer
358
views
Difference between parallel transport and ambient projection
Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
7
votes
3
answers
1k
views
Conjecture on the existence of centrosymmetric Hadamard matrices
I work with centrosymmetric matrices and recently have started exploring the question of the existence of centrosymmetric Hadamard matrices.
Definition: An $n \times m$ matrix $A = (a_{i,j})$ is ...
11
votes
5
answers
1k
views
Which directed graphs have a normal adjacency matrix?
I am working on a problem in matrix analysis and I am looking for certain types of normal matrices. I suspect that these "special" normal matrices arise as adjacency matrices of certain graphs. My ...
7
votes
2
answers
735
views
An extension of the Golden-Thompson inequality
For three symmetric positive semidefinite matrices $A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases:
$$ \operatorname{tr} \left( A e^{B+C} \right) \leq \...
0
votes
0
answers
36
views
Does this recurent matrix sequence admit an explicit writing?
I have sequence defined by : 𝐏(n+1)=(𝐈−(Ф.𝐏(n).Ф′+𝐐).𝐇′.(𝐇.(Ф.𝐏(n).Ф′+𝐐).𝐇′+𝐑)^(−𝟏).𝐇).(Ф.𝐏(n).Ф′ +𝐐)
Where :
P(n), Q, R are square, NxN, symmetric, positive semidefinite.
R is square, ...
6
votes
1
answer
387
views
Non-diagonalizability of the adjacency matrix of a directed graph
Let $G$ be a directed graph with no multiple edges or loops and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed ...
5
votes
1
answer
261
views
Bounds on number of "non-metric" entries in matrices
Question:
what upper bounds are known on the number of non-metric entries of finite dimensional square matrices $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with strictly positive off-diagonal elements $...