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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 ...

-1
votes
0answers
23 views

Tail estimate of a random matrix

How to estimate a tail of the random matrix if coordinates of each entry is known?
0
votes
0answers
26 views

Problem of expressing a two matrices product.

I think need a different method of matrix multiplication here: Suppose I have two matrices: $A = (a_{ij}) \in \mathbb{R}^{m \times m}$ and a partitioned matrix $B = (B_{ij}) \in \mathbb{R}^{mp \times ...
-2
votes
1answer
82 views

What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [on hold]

Suppose we have the following symmetric matrix. $$A = \sigma^2 I + u u^T$$ What can we say about the eigendecomposition of $A$?
6
votes
0answers
148 views

A linear algebra problem in positive characteristic (2)

Let $A$ be an $n\times n$ symmetric matrix with $0,1$ entries with all diagonal entries equal to $1$. Suppose $p>2$ an arbitrary prime number. Does always there exists $x \in (\mathbb{Z}/p\mathbb{Z}...
-1
votes
0answers
43 views

Let $M,\,K$ are skew-symmetric matrices then prove that $x_1^TMx_2=-y^T_1My_2$ and $x^T_1Kx_2=-y^T_1Ky_2.$ [on hold]

Let $M$ and $K$ are skew symmetric matrices. Suppose $\alpha Mx_i-\beta My_i+Kx_i=0,$ $\beta Mx_i+\alpha My_i+Ky_i=0$ for $i=1,2;$ where $\alpha,\beta$ are non-zero real number and $x_i,\,y_i$ are ...
17
votes
1answer
994 views

A linear algebra problem in positive characteristic

Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
2
votes
2answers
132 views

Calculate percentage of symmetry of a given matrix

Is it possible to calculate a percentage that quantifies how symmetric a given matrix is? For example, even if a given matrix is not symmetric, instead of just classifying it as "not symmetric", one ...
1
vote
0answers
54 views

Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$. I am interested in evaluating the ...
2
votes
2answers
208 views

Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently? $$(D + ADA^T)^{-1}$$ Hints or references are ...
-1
votes
0answers
41 views

Sparse matrix computational difficulties

I am a computer science student. But I start writing here because my problem is not code related. It is purely computational related. I am trying to calculated inverse of a matrix. Of course, this is ...
0
votes
0answers
141 views

How do we compute the trace over the matrix logarithm $\log((\sigma_2 \otimes I_{n/2})^T\cdot\Omega)$?

How do we explicitly compute the curvature form $\Omega$ of the Levi-Civita connection $\nabla^{L.C.}$ for the $n$-sphere $S^n$? Thus, how do we calculate the trace over the matrix logarithm $\log(...
4
votes
1answer
108 views

Smith normal form and affine buildings

In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...
1
vote
0answers
97 views

Sufficient conditions for all eigenvalues simple in stochastic matrix

The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple. ...
3
votes
1answer
206 views

Approximating the expectation of a matrix inverse

Let $$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$ where $A$ is a given $n \times m$ matrix (where $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
0
votes
0answers
94 views

Closed-form solution about a matrix factorization problem?

I am doubting some equations in paper "Multi-View Learning With Incomplete Views". The paper could be found here. The problem is related to Eq.(2) and Eq.(4b) in this paper. I summarize them here (...
4
votes
1answer
106 views

Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
5
votes
1answer
116 views

Numerical minimization spectral norm under diagonal similarity

This question is a follow up. Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \inf_{D} \lVert D^{-1} A D\...
5
votes
2answers
198 views

Minimize spectral norm under diagonal similarity

Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \min_{D} \lVert D^{-1} A D\rVert_2, $$ where $D$ ...
0
votes
0answers
80 views

norm and conorm of elliptic cocycle be different

Let $(M,\mathcal{B},\mu)$ be a probability space and $f:M \rightarrow M$ be a measure preserving map.Let $A:M \rightarrow SL(2,\mathcal{R})$be a measurable function with value invertiable $2\times2$...
2
votes
1answer
78 views

Symmetric orthogonal matrices with constant diagonal entries

$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\...
6
votes
1answer
333 views

A question on symmetric matrices

$\newcommand{\R}{\mathbb{R}}$ The question is Is there a constructive (say, parametric) description of the set (say $M_n$) of all symmetric matrices $A\in\R^{n\times n}$ such that all the diagonal ...
1
vote
0answers
49 views

Hubbard–Stratonovich for matrices

Can somebody share a link to papers( books) where this fornula was deduced? $e^{\frac{1}{2}\sum_{ij}K_{ij}s_is_j} =\left(\frac{\det{K}}{(2\pi)^N} \right)^{1/2} \int^{\infty}_{-\infty}...\int^{\infty}_{...
1
vote
0answers
70 views

Decomposition of Determinant of Sub-Matrices of a Matrix

Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means $$ {\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...
1
vote
0answers
58 views

Decomposition of a Matrix by Sparse Matrices

Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$. $\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...
1
vote
0answers
25 views

find linear approximation of non-linear matrix transform [closed]

I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists ...
4
votes
0answers
45 views

Rank of binary matrix related to the number of positive squarefree integers less than $n$

I posted this question at the Mathematics SE, but received no response there so I am posting it here. The following fact is stated in the comments-section of sequence A013928 in the OEIS. Let $C$ ...
2
votes
0answers
66 views

Characterizing a subclass of row-orthogonal matrices

Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,...
1
vote
1answer
189 views

Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset $$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$ is a manifold of dimension $2n(2r)-(...
2
votes
1answer
123 views

When does a row standardized adjacency matrix have a real spectrum?

A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
6
votes
1answer
179 views

Is this generalization of the Hopf map for quadratic field extensions surjective?

Let $k$ be a field, and let $L$ be a quadratic extension of $k$. Denote by $\sigma$ the non-trivial element of $\operatorname{Gal}(L/k)$. Let $M_2(L)$ be the vector space over $L$ of two-by-two ...
4
votes
1answer
125 views

Positive definite matrices diagonalised by orthogonal matrices that are also involutions

Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$. I want to know when this matrix is also an involution, i.e., $P^2 = I$. If there is any ...
1
vote
1answer
86 views

Equivalence of matrices over a vector space

I'm trying to characterize equivalence classes of matrices over a vector space. Specifically, let $V$ be a vector space over a field $K$, let $M \in M_n(V)$ be an $n \times n$ matrix with entries in $...
5
votes
1answer
125 views

Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$ $$ A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}. $$ I'm interested in finding a ...
9
votes
0answers
138 views

What are the periodic Dyck paths?

Let $C$ be the Cartan matrix of a finite dimensional algebra $A$ with finite global dimension, then the Coxeter matrix is defined as $M=-C^{-1}C^T$. $A$ is called periodic in case $M^k=id$ for some $k ...
0
votes
1answer
109 views

Cholesky decomposition – non-positive definite matrix

In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
0
votes
0answers
49 views

Numerical error on the spectrum of a matrix

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
3
votes
0answers
147 views

How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎ $$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^...
0
votes
0answers
46 views

How to bound the expectation of the power of sum of independent matrices?

Suppose I have $n$ independent zero mean random matrices $\{\Lambda_{i}\}_{1\leq i\leq n}$, where each $\Lambda_{i}\in \mathbb{R}^{n\times n}$ is symmetric. Also denote $E := \sum_{i=1}^{n} \Lambda_{i}...
1
vote
0answers
41 views

Is it possible to compute a valid Laplacian matrix from an effective resistance matrix?

I am wondering whether it is possible to retrieve a node-admittance matrix $G$ (also called Laplacian matrix) in a purely resistive network composed of nets $\{1, \dots, i, \dots, j, \dots, n\}$, from ...
3
votes
0answers
69 views

Group generated by symmetric shears

Consider the multiplicative group generated by matrices of the form $$ \begin{bmatrix} {1} & { 0} & { c_1} & {c_3} \\ {0} & {1} & {c_3} & {c_2} \\ {0} & {0} &...
6
votes
1answer
224 views

An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...
1
vote
1answer
78 views

On ranks of matrices with tensor structure

Fix two $2^t$ length vector of form $p=\begin{bmatrix}u_1&v_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}u_t&v_t\end{bmatrix}$ and $r=\begin{bmatrix}w_1&z_1\end{bmatrix}\otimes\dots\...
3
votes
0answers
63 views

Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
10
votes
0answers
143 views

Fast computation of matrix product $AXA^T$ with fixed $A$?

Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
2
votes
1answer
62 views

Maximise singular value decay by sparse matrix approximation

I have a large matrix $A \in \mathbb{R}^{n \times m}$ and would like to subtract a sparse matrix $B \in \mathbb{R}^{n \times m}$ with less than $c (n+m)$ non-zero entries, where $c > 0$ is a ...
4
votes
0answers
86 views

Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

I am ready to agree beforehand that this looks more like a math.SE question. I posted it there a week ago without any feedback (except for 27 views and 2 upvotes). Besides, I really need an answer. ...
15
votes
2answers
662 views

Is this lower bound for a norm of some complex matrices true?

Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$. I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(...
0
votes
1answer
94 views

prove the singularity of a matrix as solution of a non-linear equation

Let $B$ ($n \times n$) and $R$ ($m \times m$) be two square matrix with $n>m>0$ who satisfie: $B=(I-KH)B(I-KH)^T+K RK^T$ with $K=BH^T(HBH^T+R)^{-1}$ and $rank(H)=m$ I would like to prove $...
1
vote
1answer
113 views

Singular values of Hadamard Product

For two Matrices $A,B \in \mathbb{R}^{m \times n}$ the Hadamard Product is defined as $(A \circ B)_{i,j} = A_{i,j}B_{i,j}$. For a proof of convergene I require an upper (and ideally a lower) bound on ...
12
votes
1answer
349 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...