Questions tagged [matrix-theory]

Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.

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Nontrivial lower bound on the sum of matrix norms

Let $X, V\in\mathbb{R}^{n\times r}$ such that $X^\top V$ is symmetric. The central quantity I care about is \begin{equation} \|XV^\top\|_{F}^2+\|X^\top V\|_{F}^2 +[\text{Tr}(X^\top V)]^2. \end{...
Wuchen's user avatar
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10 votes
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Matrix identities in two variables

The famous Amitsur-Levitzki Theorem states that the algebra $M_n(\mathbb C)$ satisfies no polynomial identity of degree less than $2n$ and it satisfies $$p(x_1,\dots,x_{2n}) = \sum_{\sigma \in \frak ...
Andreas Thom's user avatar
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-1 votes
1 answer
303 views

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 ...
Turbo's user avatar
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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 ...
Ali Taghavi's user avatar
2 votes
0 answers
171 views

Bounding the distance between two matrix power sequences

Let $A,B$ be Hermitian matrices so that $0 \le A,B < I$ and also $(1-\varepsilon)(I-B)\le I - A \le (1+\varepsilon)(I-B)$. For every $t \in \mathbb{N}$, consider the matrix $A_{t} = \sum_{i=0}^{t}...
Daniel86's user avatar
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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 ...
Wuchen's user avatar
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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? ...
Abhishek Halder's user avatar
1 vote
1 answer
125 views

Redistribute diagonal entries of a matrix

Let $d = (d_1,...,d_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with: $V$ ...
Yair Daon's user avatar
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1 vote
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Fullrankness of sum of time shifts

I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now. Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
nahila's user avatar
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2 answers
475 views

Does $R$ is Dedekind-finite imply $\mathbb{M}_n(R)$ is Dedekind-finite

Following Lam's notation, a ring (with identity) $R$ is called Dedekind-finite if $ab=1\iff ba=1$ in $R$. There are a lot of result about left invertible implies right invertible. But the results all ...
Cubic Bear's user avatar
3 votes
1 answer
203 views

Better name for “vec transposition permutation matrix”?

Let the operator vec($A$) unroll all the elements of $A$ into a single column vector in column-major order. Then, the elements of vec($A^T$) are a permutation of the elements of vec($A$). If I want to ...
Niclas Börlin's user avatar
1 vote
0 answers
178 views

Maximum singular value of sum of an Hermitian and an anti-Hermitian matrix

Let $H$ be an $n\times n$ Hermitian matrix and $A$ an $n\times n$ anti-Hermitian matrix, i.e. $H^\dagger = H$, $A^\dagger = -A$. Consider their sum $S= H+A$. Let $\{\sigma_i(S)\}_{i=1,\dots,r}$ denote ...
usr73617381's user avatar
1 vote
1 answer
215 views

zero patterns of M-matrices and their invereses

Suppose that the matrix $M$ has non-positive off-diagonal elements. The matrix $M$ is said to be an M-matrix if $M$ is non-singular and each entry of $C:=M^{−1}$ is non-negative. I think I've proved ...
Giuliano Basso's user avatar
2 votes
2 answers
121 views

Behavior of matrix rank under thresholding of its elements

Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is ...
gradstudent's user avatar
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3 votes
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Eigenvalues of block-hermitian matrices with zero diagonal blocks

I have a matrix of the form $$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$ where $C$ is not necessarily hermitian. In general, can we say anything about the ...
Unwieldy Bob's user avatar
2 votes
0 answers
164 views

Lower bound for the sum of cosines between singular vectors of diagonally dominant matrices

Let $A \in \mathbb{R}^{n \times n}$ be a nonsymmetric diagonally dominant matrix with $a_{ij} < 0$ $\forall i \ne j$ and $a_{ii}>0$. Let the singular value decomposition of $A$ be $A=U \Sigma V^...
Astor's user avatar
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7 votes
1 answer
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When does the determinant distribute over addition?

When does $\det(A+B)=\det(A)+\det(B)$ hold? I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
Turbo's user avatar
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17 votes
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Finding the nearest matrix with real eigenvalues

In this thread on MATLAB Central, I found a discussion on finding the nearest matrix with real eigenvalues. The first hypothesis was to simply truncate the complex part of the eigenvalues. So, given ...
Andrea F.'s user avatar
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6 votes
2 answers
248 views

A correlation matrix problem

I have a linear algebraic question about an arbitrary correlation matrix. If $$ \lambda_{\min} \begin{pmatrix} 1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_3 \\ \rho_2 & \rho_3 & ...
Nikolayevich's user avatar
1 vote
0 answers
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relationships between $AA^T$ and $[(I-A)(I-A)^T]^{-1}$ with $A$ being strictly lower triangular

I have a matrix $A$ which is strictly lower triangular. Now, I am trying to find some general statements/relationships of following matrices $U,D,V,K$ defined as: $AA^T=VKV^H$, $[(I-A)(I-A)^T]^{-1}=...
user3093264's user avatar
3 votes
0 answers
55 views

Equivalence Classes of a Subgroup of Similarity Transformations

Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices $$ \begin{bmatrix} A & B\\ C & D\\ ...
JMJ's user avatar
  • 263
8 votes
1 answer
460 views

Determinants (and traces) of linear maps of matrices

Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in ...
M.G.'s user avatar
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2 votes
2 answers
117 views

Powers of small square matrices over the Laurent polynomial ring with integer coefficients

I'm trying to calculate the powers of a 2 by 2 matrix with entries in $\mathbb{Z} \left[ t,t^{-1} \right]$. The matrix is \begin{bmatrix} 0 & 1 \\ 1 & t \end{bmatrix} I tought of writing my ...
Mohammed Sabak's user avatar
1 vote
1 answer
577 views

Can I modify the singular values of a matrix in order to get a negative eigenvalue?

Let $A \in \mathbb{R}^{n \times n}$ be a real nonsymmetric matrix with eigenvalues $\left\{\lambda_i : i=1..n\right\}$ with positive real part $\Re(\lambda_i) > 0$ $\forall i=1..n$ Let $A=U\Sigma ...
Astor's user avatar
  • 323
2 votes
1 answer
340 views

Simultaneous triangularization of two diagonal matrices and a symmetric matrix

I have the following quadratic eigenvalue problem: $\det(\lambda^2M + \lambda D + J)=0$ where, M and D are both $n \times n$ real diagonal matrix with positive diagonal entries; $J$ is 1) a ...
Pascal's user avatar
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1 vote
0 answers
76 views

When is $F(X)BF(X)$ operator monotone, if $F(X)$ is operator monotone?

Let $\Omega_{n}$ denote the cone of $n\times n$ real symmetric positive definite matrices, and consider $F:\Omega_{n} \mapsto \Omega_{n}$. For $X,Y \in \Omega_{n}$, the matrix valued function $F(\cdot)...
Abhishek Halder's user avatar
8 votes
1 answer
354 views

Rank of a combinatorial matrix

For any $2n$ with $n\in\mathbb{N}$, we conjecture the matrix $A\in\mathcal{M}_{3n\times 2n}$ to have rank $\lceil{\frac{5n-1}{3}}\rceil$ where $$A=\left(\begin{array}{cccccc} 0 & & & &...
lsilverstein's user avatar
3 votes
2 answers
2k views

Generalized Hölder's inequality for operator (subordinate) norms

While perusing the Matrix norms section of Wikipedia, I came across this generalized version of Holder's inequality. $$ \|A\|_2^2 \leq \|A \|_1 \|A \|_\infty\,, $$ where, $$ \|A \|_p = \max_{\|x\|_p ...
GraspIt's user avatar
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2 votes
0 answers
1k views

Derivative of determinant of (parameterized) non-invertible matrix [closed]

Consider a function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}, t \rightarrow A(t)$. I want to compute the derivative of the determinant $$\frac{d}{dt} \det(A(t)) \; .$$ Suppose $A(t)$ is ...
madison54's user avatar
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10 votes
1 answer
577 views

is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?

For two regular graphs $G$ and $H$, it is possible for them to share the same adjacent spectrum and the same laplacian spectrum. While, on the other hand, is it possible to have two non-regular graphs ...
josephS's user avatar
  • 103
1 vote
0 answers
119 views

An analogue of Hermitian matrix - does it exist?

Let $k$ be any field and $R\subseteq M_s(k)$ be a subring of $s\times s$ matrices over $k$. Identify $k$ with the scalar matrices, so that $k\subseteq R$. Let $A\in M_n(R)$ be an $n\times n$ matrix. ...
Adam Przeździecki's user avatar
3 votes
0 answers
447 views

"Natural" ways of interpolating unitary matrices

Given two unitary matrices $A$ and $B$, that are "near" each other in some sense (perhaps $\left\lVert A-B\right\rVert <\epsilon$ for some norm, what are some sensible ways to interpolate between ...
Victor Liu's user avatar
6 votes
0 answers
321 views

Exact determinant of a Cauchy-like matrix

Question. Is there a closed-form expression for the determinant of a $n \times n$ matrix $A$ with entries $$ A_{i,j} = \frac{1 - \delta_{i, j}}{z_i - z_j}, \qquad 1\leq i, j\leq n, $$ where $z_i$ ...
Jeannette's user avatar
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1 vote
0 answers
125 views

Matrix majorization when a diagonal matrix is multiplied from right and left

Let $D_1$ and $D_2$ be two diagonal matrices such that $D_1^2+D_2^2=I$ (identity matrix). Suppose matrix $A$ majorizes matrix $B$. Can we show that matrix $A$ majorizes matrix $D_1 A D_1 + D_2 B D_2$? ...
Soheil Feizi's user avatar
2 votes
1 answer
422 views

equality between the ratio trace and the determinant ratio

I have encountered the following equality $\arg\max_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{A}\boldsymbol{S}\left(\boldsymbol{S}^{H}\...
Student88's user avatar
  • 503
3 votes
1 answer
95 views

Isomorphism concerning $Soc(M_n(R))$

It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
karparvar's user avatar
  • 323
4 votes
2 answers
309 views

Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$

For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix. I would like to solve the following equation for the ...
Abhishek Halder's user avatar
4 votes
1 answer
2k views

Diagonalization of real symmetric matrices with symplectic matrices

Consider the following real symmetric matrix $M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$ Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$...
fagd's user avatar
  • 41
1 vote
0 answers
242 views

Product of exponentials of matrices

Let $A$ and $B$ be commuting $n\times n$ positive definite complex matrices and let $C$,$D$ be other complex matrices. I wish to think of $C,D$ as small and $A,B$ as any size. Suppose we are looking ...
JRoss's user avatar
  • 270
4 votes
0 answers
112 views

Inducing surjections on $GL_n(-)$?

Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $...
BillScroggs's user avatar
1 vote
1 answer
100 views

Under what condition can any $X\in GL_2(R)$ be reduced to a triangular matrix?

Suppose $R$ is a (possibly noncommutative) ring. I was thinking of $R=S[x_1,\ldots,x_n]$ or $R=S[x_1,x_1^{-1},\ldots,x_n,x_n^{-1}]$ for $S$ some (possibly noncommutative) ring. Now, let $GL_2(R)$ be ...
Sam Williams's user avatar
5 votes
2 answers
2k views

Decomposing a matrix into a product of sparse matrices

How to study the decomposition of a square matrix into a product of sparse matrices? There are no restrictions on the number of matrices in the product, but the fewer the better.
unknown's user avatar
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-1 votes
1 answer
61 views

Finding a matrix with shared eigen vectors with a given matrix [closed]

If I have a known matrix A, is there a method to find a matrix B that share all the eigen vectors of Matrix A?
Weera's user avatar
  • 9
3 votes
1 answer
3k views

Solving a vector of quadratic equations

I have a system of $n \times 1$ equations $$ 0 = A\,vec(xx^t) + B x + C $$ where $x$ is a $n \times 1$ vector of unknowns $x^t$ means transpose $vec$ means $xx^t$ has been vectorized so has dimension ...
Oliver's user avatar
  • 43
3 votes
2 answers
131 views

Vanishing zeroes in matrix powers

For a matrix $M\in\mathbb{R}^{n\times n}_{\geq 0}$ with nonnegative entries, we define $m$ as the smallest positive integer such that all the entries of $M^m$ are strictly positive (if there is one). ...
Hauke Reddmann's user avatar
1 vote
0 answers
115 views

Low-rank approximation of sub-sampled matrix

Considering a large data matrix $X$ with zero-centered columns that is assumed to be approximately low-rank, it is common to do PCA and project the data onto the top few principal components, and use ...
user310374's user avatar
10 votes
2 answers
1k views

When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?

Suppose $A_k>0$ (which means they are positive definitive square $n\times n$-matrices with $n>1$). If $\sum_{k=1}^\infty A_k$ exists, then $\sum_{k=1}^\infty \|A_k\| < +\infty$, Where $\|A\|=\...
Xifeng Su's user avatar
  • 173
2 votes
1 answer
293 views

Submatrix with small sum of elements

Let $A$ be an $n \times n$ matrix, for which I know the size of the sum of all its entries. Now I want to select an $m \times m$-submatrix, whose sum of entries is as small as possible. Is there any ...
Kurisuto Asutora's user avatar
4 votes
0 answers
433 views

A sum of Ramanujan sums

I have the following question about Ramanujan sums. (All vectors and matrices here will be understood to have integer entries.) Let $X_q=\{ (x_1,...,x_R)|1\leq x_i\leq q\} $ and let, for any $R\...
tomos's user avatar
  • 1,096
5 votes
4 answers
2k views

Differentiability of eigenvalue and eigenvector on the non-simple case

Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
Shake Baby's user avatar
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