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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2
votes
1answer
154 views

When is there a solution to these coupled eigenvalue equations?

I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
15
votes
4answers
986 views

Condition for two matrices to share at least one eigenvector?

Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so ...
0
votes
0answers
137 views

What matrix operation is this?

Does this matrix operation have a name? Let $A$ and $B$ be two $2^m$ by $2^n$ binary matrices. Then the element in the uth row and wth column of the new matrix is $ \sum_{v=0}^{2^n-1} a_{u, v ...
2
votes
1answer
278 views

Some puzzles about the three conditions in a paper of D.Berend

Recently, I am reading a paper titled "multi-invariant sets on tori" by D.Berend. I am puzzled by the three necessary and sufficient conditions given there. Could you provide me with some concrete ...
7
votes
2answers
188 views

Sum of Difference of anti-diagonal matrix elements

Let $A \in \mathbb{R}^{n \times n}$, with elements $a_{ij}$ What conditions on $A$ are required for the following to be true? There exists some vector $x \in \mathbb{R}^n_+$, $x \neq 0$ such that ...
3
votes
1answer
118 views

On matrices conjugated in a faithful representation

Let $k$ an algebraically closed field. Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group. Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...
2
votes
0answers
82 views

integral stable conjugacy classes

Let $G$ be a semisimple simply connected group over $k$ algebraically closed field . Let $\gamma,\gamma'\in G(k[[\pi]])$ that are generically regular semisimple on $G(k((\pi)))$. We assume that ...
4
votes
0answers
166 views

How to find the unitary matrices in this exponential matrix representation

In the following post Representing a product of matrix exponentials as the exponential of a sum there is a statement regarding the result of the multiplication of two matrix exponentials: if $A$ and ...
5
votes
0answers
196 views

Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for ...
2
votes
1answer
126 views

Number rank-k 0-1 matrices (characteristic 0)

What is the number of $n\times n$ 0/1-matrices with rank $k$? (The rank is taken over the rationals.)
4
votes
1answer
198 views

Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)

Given correlation matrix $B$ (positive semi-definite with ones in the diagonal). 1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$. 2)Find the correlation matrix $A$ which ...
-2
votes
1answer
313 views

For a ring $A$, is $A$ Morita equivalent to $M_\infty(A)$? [closed]

Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if ...
4
votes
0answers
104 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...
2
votes
0answers
142 views

Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$. The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
2
votes
3answers
183 views

On matrices in linear forms with vanishing determinant

This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought. Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...
12
votes
1answer
386 views

Does the Hasse principle hold for the square root problem on symmetric matrices?

Let $S$ be an $n \times n$ symmetric matrix with rational entries. It is known that the equation $XX^T=S$ has a solution in $\text{M}_n(\mathbb{Q})$ if and only if it has a solution in ...
4
votes
1answer
290 views

Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as ...
3
votes
0answers
83 views

Large prime divisors in entries of matrix powers

Are any examples known of an integer matrix $A$, such that the largest prime divisor of some specified entry of $A^n$ grows exponentially in $n$? How about where it just grows strictly faster than any ...
0
votes
0answers
98 views

a very elementary question on the conjugated matrices

Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero . We suppose that they have the same ...
0
votes
1answer
342 views

eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
3
votes
0answers
198 views

An optimization problem over real symmetric matrices

Given an $n\times s$ matrix $P$ of positive real numbers and $T\geq n$, find (either by a formula or an algorithm) the real symmetric $n\times n$ Z-matrix $A$ which maximizes $\min\limits_{1\leq ...
7
votes
2answers
339 views

Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set $$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$ Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by ...
0
votes
1answer
113 views

Number of minimal left ideals

Is there any way to compute the number of minimal left ideals of $M_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?
5
votes
0answers
87 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. ...
1
vote
1answer
98 views

Finding the most compact representation of a vector in an “overdetermined base”

I want to find the most compact representation of a vector as a linear combination of a set of vectors B. B has more elements (on purpose) that is needs to have to describe the subspace. For example ...
3
votes
2answers
717 views

Sparse approximation of the inverse of a sparse matrix

Is it possible to approximate an inverse of a sparse matrix with a sparse matrix? The problem comes up in numerical non-linear quasi-Newton optimization: given a sparse Hessian a good starting point ...
3
votes
0answers
120 views

A similarity problem over matrices over Gaussian integers

Let $R = \mathbb Z[\sqrt{-1}]$ and $$\Omega = \{X \in GL_4(R) : X \overline X = I_4 \text{ or } -I_4 \},$$ where $\overline X$ is the complex conjugate matrix of $X$. Two matrices $A, B \in \Omega ...
8
votes
2answers
735 views

How to transform matrix to this form by unitary transformation?

Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e. $$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$ Then is ...
4
votes
0answers
239 views

How to calculate the inverse of a matrix with block Kronecker structure?

I have a Hermitian matrix $A$ with the following block Kronecker structure, $A = \begin{pmatrix} R_0 \bigotimes S_0 & R_1 \bigotimes S_1\\ R_1^H \bigotimes S_1^H & R_0 \bigotimes S_0 ...
2
votes
0answers
120 views

Random variable matrix exponential

I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here. What ...
1
vote
3answers
183 views

Geometric means of matrices beyond the positive definite cone

Recently a lot of work has been done on geometric means of positive definite matrices (see here and here for example). Has anyone extended this concept to larger sets of matrices (copositive, for ...
1
vote
1answer
328 views

Iwasawa Decomposition for Matrices [closed]

I was asked to prove that if $$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$ denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...
4
votes
1answer
227 views

A complex sequence with positive values

Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d ...
5
votes
1answer
272 views

a question of ranks of matrices

Let $S=\{A_{1}, ..., A_{m}\}$ be a set of $n \times n$ symmetric matrices and $m>n$, the rank $r(A_{i})=1$ for each $i$. Suppose that for any $m-1$ matrices $\{A_{i_{1}},...,A_{i_{m-1}}\}$ in $S$, ...
3
votes
2answers
335 views

What is known about the spectrum of a Cauchy matrix?

Math people: A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form $a_{i,j} = \frac{1}{x_i-y_j}$, with $x_i \neq y_j$ for all $(i, j)$, and the $x_i$'s and $y_i$'s belong to a ...
4
votes
1answer
196 views

Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix

Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows. For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...
5
votes
0answers
158 views

Bound on number of multiplications required to generate a matrix algebra from generators?

I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all? Suppose you have ...
3
votes
1answer
151 views

Explicit formula for an LMI solution

Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil): $$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$ (The notation $X \succeq Y$ means that ...
3
votes
1answer
301 views

numerical range of a column-zero-sum matrix

I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. ...
3
votes
3answers
335 views

A NICE necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure!

Let $$ A = \begin{pmatrix} \sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\ -a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\ \vdots & \vdots & \ddots & ...
18
votes
3answers
851 views

looking for proof or partial proof of determinant conjecture

Math people: I am looking for a proof of a conjecture I made. I need to give two definitions. For distinct real numbers $x_1, x_2, \ldots, x_k$, define $\sigma(x_1, x_2, \ldots, x_k) =1$ if $(x_1, ...
8
votes
1answer
136 views

Growth of powers of non-negative integer matrices

In what I am currently doing, there naturally appears the following question: let $A$ be a square matrix with non-negative integer entries. Let $a_n$ be the sum of all entries of $A^n$. Question: How ...
3
votes
1answer
122 views

Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
2
votes
2answers
149 views

Is my use of the eigendecomposition correct here?

I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
7
votes
1answer
252 views

Combinatorics of resultants

This is a crosspost of http://math.stackexchange.com/questions/446470/combinatorics-of-resultants which received no answer. [EDIT: I deleted the initial copy of the question on MathSE]. Let ...
4
votes
2answers
115 views

A certain type of constrained Rayleigh-Ritz ratio

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem \begin{align} \max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\ ...
10
votes
1answer
358 views

A simple proof for a theorem of Szekeres and Turán

Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
3
votes
2answers
587 views

A problem about Determinant of sum of permutation matrices

Let $w_1$ and $w_2$ be two permutations of $\{1, \cdots , k\}$ such that for all $1\leq i \leq k$, $w_1(i)\neq w_2(i)$. Let $m$ and $n$ be two relatively prime integers. Then is there exist two ...
6
votes
1answer
330 views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as $H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$ What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$? For example, ...
2
votes
1answer
127 views

Finding null-homologous curves via the matrix equation $AB^iC^jx=0$

Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves ...