# 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 ...

2,044 questions

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### Simple way to generate (or characterize) real $N \times N$ matrices $K$ satisfying $(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \subseteq [\![N]\!]$

Let $N$ be a large positive integer.
Question
What is a simple way to generate (or characterize) real $N \times N$ matrices $K$ such that
$$
(-1)^{|A|}\det(K-I_A) \ge 0,\;\forall A \subseteq [\![N]\...

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15 views

### Matrix Product Chain Representation of an Addition Chain

I am looking for references to anything interesting thats know about matrix product chains that take the vector $\{1\}$ to another vector $\{n\}$ (the end result of an addition chain). Each matrix ...

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14 views

### Matrix calculation of collision in 3D-plane [on hold]

Given a sphere(S) with the center a(2,0,0) with the radius of √2. A spherical object flying with the direction R=(-1,-2,0) hits (S) at point t=(3,1,0) and is reflected on it. Compute the direction of ...

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40 views

### On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...

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68 views

### A question related to (random) matrix factorization

Let $\mathbf{B}$ be an $m\times r$ binary matrix taking values in $\{0,1\}$, and assume that $m\geq r$. I am wondering what can be said about the recovery of $\mathbf{B}$ from $\mathbf{B}\mathbf{B}^T$....

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26 views

### A $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...

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87 views

### Hadamard matrix research [closed]

I need to find out whether something new has happened in the study of Hadamard matrices for the last 10 years.
If someone was engaged in matrices, can you recommend literature?
K. Horadam is the ...

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299 views

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### Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?

For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...

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62 views

### k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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75 views

### Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...

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100 views

### Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic

This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller.
Let $\mathbb{N}$ denote the set of positive integers and for $n\in\...

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223 views

### How to verify the characteristic polynomial? [closed]

I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?

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88 views

### A generalization of matrix minors to non-integer values

I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$
One approach I thought of was to use the fact that the $k$-minors are (...

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35 views

### Computing with a vector subspace equipped with a prescribed basis over finite fields

Let $U$ be a subspace of the finite dimensional vector space $V$ over a field $\mathbb{k}$. Let $B_V$ and $B_U$ be fixed bases for $V$ and $U$ respectively. Let $u \in U$ and let's give ourselves $[u]...

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59 views

### matching two positive-semidefinite matrices

Let $M_1$ and $M_2$ be two real positive-semidefinite matrices. Is there any algorithm to compute a permutation matrix $P$ to minimize $\| M_1-PM_2P^T \|_F^2$ or equivalently to maximize $trace(...

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76 views

### Generic deformation of matrix

Let $A(x)$ be a $m \times n$ matrix, whose entries are real polynomials $f_{i,j}:\mathbb{R}^S \to \mathbb{R}$. Denote the ith row by $f_i$ And let $rk:\mathbb{R}^S \to \mathbb{N}$ be the rank function ...

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**1**answer

308 views

### Decide if a matrix is transposable

A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...

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**1**answer

82 views

### Spin groups in terms of matrices and/or linear operators

Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...

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197 views

### Determinant of a block matrix with many $-1$'s

For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...

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**1**answer

235 views

### Determinant of a matrix filled with elements of the Thue–Morse sequence

Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be ...

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100 views

### Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...

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71 views

### Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?

Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...

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**1**answer

234 views

### Is there a formula for the determinant of a block matrix of this kind?

I am looking for an expression that gives the determinant of a matrix of the form
\begin{bmatrix} A & B & 0 & \dots & 0 & C \\
B & A & B & & 0 & 0 \\
0 & ...

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174 views

### Making binary matrix positive semidefinite by switching signs

Let $A \in \{\pm 1\}^{n \times n}$ be a symmetric matrix whose diagonal entries are $+1$. Let $f(A)$ be the smallest number of signs we need to change in $A$ so that it becomes positive semidefinite (...

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203 views

### Legendre's three-square theorem and squared norm of integer matrices

Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...

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51 views

### Can a rational symmetric matrix $A$ be diagonalized as $A = P \Lambda S$ for some $P, S $ in the general linear group?

Let $A$ be a $3 \times 3$ symmetric matrix with rational entries. Does there exists a unique pair of matrices $P, S \in \text{GL}_3(\mathbb{Z} )$ depending on $A$ such that $A = P \Lambda S$, where $\...

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432 views

### Reference request: Oldest linear algebra books with exercises?

Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....

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232 views

### A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...

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101 views

### Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...

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51 views

### How likely is a matrix with exactly $n$ number $1$s per row to avoid a large wide empty submatrix?

Consider the finite collection $M(N,n)$ of all $N \times N$ matrices with exactly $n$ entries per row equal to $1$ and all other entries equal to zero $0$.
By an $a \times b$ submatrix of $M$ we ...

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80 views

### Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...

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83 views

### Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...

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**1**answer

82 views

### Matrix equation of the form $C A C^\intercal = D$

Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...

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**1**answer

171 views

### Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?

Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...

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**1**answer

171 views

### Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...

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76 views

### Tiling with Horn's polytopes

Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...

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83 views

### another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$
In 1984 S.D.Berman, a Soviet mathematician, ...

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138 views

### Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?

Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...

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148 views

### Generalizing Polar Decomposition of Matrices

I am trying to find a certain proof of polar decomposition of complex matrices which I think should exist more generally for a certain class of Lie groups. Recall that the polar decomposition of a ...

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12 views

### Every singular DNN realization of $G$ is completely positive implies

DNN denotes doubly non-negative matrices(both entry wise non-negative and is positive semi-definite). Let $G$ be a Graph.
The following two are equivalent:
(a) Every non-singular DNN realization of $...

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**1**answer

106 views

### Eigenvalues of A^T D A for positive A and diagonal D

Suppose I have a diagonal matrix $D$ whose entries are bounded in absolute value. I also have a matrix $A$ that is positive (entry-wise, so $A_{ij} > 0\ \forall\ i,j$): one can assume that the ...

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176 views

### The determinant of a $4\times4$ matrix associated to some specific polynomial as follow

Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...

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42 views

### What can we say about the dominant Generalized eigen vector of two matrices A and B if the dominant eigen vector of A and B are known?

I have the following problem: $\bf{a} = V_{max}(A,B)$, where $V_{max}$ refers to the dominant generalized eigen vector solution and A, B are two $N \times N$ matrices (full rank). Suppose i know the ...

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114 views

### Existence of an “almost” skew-symmetric matrix

Let $A\in\mathbb{R}^{3\times 3}$ be a matrix of the form
$$
A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ -a_{12} & a_{22} & a_{23} \\ -a_{13} & -a_{23} & a_{33} \end{bmatrix}
$...

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**1**answer

193 views

### On the existence of integer square root of a $3 \times 3$ positive definite matrix

As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique.
I have read some ...

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**0**answers

86 views

### On a matrix inequality based on the Schur-Horn theorem

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.)
Let $A_s$ denote the symmetric part of $A$...

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223 views

### A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...

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122 views

### Space of change of basis matrices between two similar matrices - how to reduce it with additional tests?

Assume we have two real symmetric $n\times n$ matrices: $A, B$. We can easily test their similarity: $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1..n$. In this case both can be rotated to the same ...

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117 views

### A “positive diagonal plus skew-symmetric” matrix decomposition

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric).
My question. Do there exist an orthogonal ...

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**1**answer

113 views

### On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...