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 ...

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4
votes
0answers
29 views

Fast matrix-vector product for structured matrices

Let $X\in\mathbb{C}^{m\times n}$ be a matrix that satisfies the Sylvester equation $$AX-XB = F,\qquad A\in\mathbb{C}^{m\times m}, \quad B\in\mathbb{C}^{n\times n},$$ where $F\in\mathbb{C}^{m\times n}$...
0
votes
1answer
68 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $0 \le a_{j,j} \le 1$ and $-1 \le ...
0
votes
0answers
19 views

Criteria for existence of stable principal submatrices of a stable matrix?

Let $A$ be an $n\times n$ real matrix. Suppose $A$ is stable, that is, all the eigenvalues of $A$ have strictly negative real part. Question: What are some results about existence of stable ...
0
votes
0answers
32 views

How to find $K,W,S$ in the Mostow decomposition theorem?

The Mostow decomposition theorem states: Let $Z$ a nonsingular complex matrix, then $Z$ can be factored as: $$Z=We^{iK}e^S$$ where: $W$ is unitary, $K$ is real and skew symmetric and $S$ is real and ...
0
votes
0answers
56 views

An idea of two sided invertible matrix [closed]

Can you suggest me a reading material that expand on the idea of two matrices that are inverse to each other but which are not square matrices? Here's what I think of, take $A$ a matrix of order $n\...
2
votes
1answer
75 views

Minimize matrix distance to tensor product

Minimize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
2
votes
0answers
26 views

Metrics on the group of unimodular polynomial matrices

The group of unimodular matrices $\mathbb{U}[s]^{n\times n}$ is given by the set of $n\times n$ square (real) matrix-valued polynomials $\mathbb{R}[s]^{n\times n}$ which admit a polynomial inverse. ...
3
votes
1answer
209 views

Lie algebra of invariant polynomials or invariant smooth functions

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb{...
2
votes
1answer
102 views

Maximize inner product of a tensor of unitary matrices

How can one maximize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$. Both the maximum value of ...
1
vote
0answers
39 views

determine the existence of positive semi-definite matrix

Given a $d\times d$ complex matrix subspace $S$, we are asking whether there is some finite integer $n$ such that there exists a non-zero positive semi-definite matrix is orthogonal to $S^{\otimes n}$....
1
vote
1answer
29 views

nonnegative solution of nonhomogeneous under-determined linear system of equations

For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such ...
1
vote
1answer
157 views

Is this a full rank matrix? [closed]

According to the answer of znt to the previous version, I revise the question as follows: Is there a real $(n-1)\times n$ matrix $A$ such that $A$ is not a full rank matrix and satisfy $a_{ii}&...
1
vote
0answers
46 views

Expected amount of linearly dependent random vectors? [closed]

Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$? EDIT: As said in the comments, I'm looking for the ...
1
vote
0answers
26 views

Tight upper bound for the degree of the entries of adjugate of polynomial matrix

(This question was originally asked at Math.SE, where it didn't receive any answers.) Let $A(x_1, \ldots, x_m)$ be a $n$ x $n$ matrix whose entries are polynomials on real variables $x_1, \ldots, ...
0
votes
0answers
27 views

convex representation of a combinatorial constraint

I have an optimization problem with a weird constraint as follows. Is it possible to express it in some ways that have convex properties: matrix $\mathbf{X}$ is either $[1 \ 0 \ 0 \ 0 \ 0\\ \ 0 \ 0 ...
0
votes
1answer
81 views

A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result: For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...
0
votes
0answers
55 views

Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...
2
votes
1answer
110 views

Carathéodory's theorem for $SO(3)$?

Let $Q \in \operatorname{Conv} SO(3)$. Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}}, R_{i} \in SO(3)$? An approximation ...
0
votes
0answers
62 views

Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$. Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is $M[i,j]={(...
7
votes
0answers
115 views

Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices. It would be great to have an ...
2
votes
0answers
60 views

How to find a closed form of following matrix's determinant [closed]

I wanna find a closed form of determinant of the following matrix $$A(n) = \begin{pmatrix} B_{1} & B_{2} & \cdots & B_{n} & 1 \\ B_{n} & B_{1} & \cdots & B_{n-1} &...
4
votes
1answer
81 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
0
votes
0answers
21 views

Number of Asymmetric, Balanced Permutation Matrices

let $\mathcal{ABP}$, the set of "asymmetric, balanced permutation matrices", be defined as the set of permutation matrices, that aren't equal their transpose but, for which the number $1$s below the ...
3
votes
0answers
29 views

Selecting columns from multiple matrices to form a well-conditioned matrix

Given multiple matrices of the same size, is there a way to select one column from each matrix to form a well-conditioned matrix? For example, given four 4-by-10 matrices A, B, C, D (real, positive, ...
0
votes
1answer
83 views

How does the rank of $C_i$ change with $i$?

Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ ...
0
votes
0answers
13 views

Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix). With some ...
0
votes
0answers
46 views

SVD alternatives for symmetric matrices

Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices $$ A = E'E $$ Practically this can be done easily using SVD ...
1
vote
2answers
69 views

How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...
0
votes
0answers
34 views

Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, $$\lvert\lambda_1-...
0
votes
1answer
69 views

Matrix norm inequality for C*-Algebras [closed]

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...
1
vote
0answers
39 views

Matrix transformation [closed]

I want to show that $(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$ is equivalent to $M_\gamma$ has no purely imaginary ...
6
votes
2answers
849 views

What is the Time Complexity of the Matrix Exponential?

While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm(). According to ...
1
vote
0answers
69 views

Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
6
votes
1answer
221 views

A sufficient condition (or not) for positive semidefiniteness of a matrix?

Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite. I do not think the ...
4
votes
1answer
98 views

Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
6
votes
1answer
235 views

coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
2
votes
1answer
160 views

The class of $(-1,0,1)$-matrix with all row sums and column sums equalling to $0$

Let $n$ be an even positive integer and $W_n$ be the class of all $n\times n$ matrices with entries from the set $\{-1,0,1\}$ satisfying all row sums and column sums are equal to $0$. For any $M\in ...
5
votes
2answers
293 views

Explicit solution to a Rayleigh quotient equation

For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method): Main equation: $$ \biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\...
4
votes
1answer
77 views

Perturbations on the pseudoinverse of a matrix

Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation $$ A_p = A + \Delta $$ is there a way to represent $$ (A_p)^{\star}= (A)^{\star} + f(\Delta) $$ where $(A_p)^{\star}$ ($(A)^{\star}...
11
votes
3answers
241 views

Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

I've read that if $M_1, \dots, M_n$ are matrices in $\mathrm{SL}(2, \mathbb{Z})$ whose product is the identity, and each is conjugate to the shear $$ \begin{pmatrix} 1 & 1 \\ 0 & 1\end{...
0
votes
1answer
88 views

Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices. I've been searching everywhere for an ...
3
votes
0answers
91 views

Finding nearest Toeplitz matrix to a given matrix

For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change. Specifically I want find the Toeplitz ...
0
votes
2answers
60 views

Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix. I've seen a similar ...
4
votes
1answer
107 views

I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?

This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated. Question I have a very ...
3
votes
1answer
141 views

Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite. $P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$, and when $A$ ...
1
vote
0answers
18 views

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...
9
votes
0answers
211 views

Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
1
vote
1answer
85 views

Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that $A=QBQ^{...
-2
votes
1answer
113 views

Eigenvalues of cyclic tridiagonal matrix [closed]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum n_i=n.$ Prove that this matrix ...
2
votes
1answer
82 views

Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question: Bounding the absolute sum of entries of the ...