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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0
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0answers
28 views

Reducing the degrees of freedom in unitary columns

Let $U = diag([U_1, U_2, ..., U_N])$ be a block-diagonal $NM \times NM$ unitary matrix, where each $U_j$ is a unitary $M \times M$ matrix. Furthermore, let $Q_e = kron(I_M, Q)$ be the Kronecker ...
7
votes
2answers
186 views

Computer Algebra Systems that support variable sized matrices

I'm familiar with sympy, the matlab symbolic package, reduce, and have tried out a few other computer algebra systems. However, as far as I can tell, none of them seem to be able to do algebra on ...
3
votes
1answer
109 views

The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$

Motivated by the following RG question we ask a related question as follows: We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes ...
8
votes
0answers
89 views

Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...
0
votes
0answers
66 views

Expected value of minimum rank of random matrices

I have $n$ random vectors ${\bf r}_i$ for $i=1,2,\dots,n$, each with dimension $1 \times m$, and $n$ random matrices ${\bf S}_i$ for $i=1,2,\dots,n$, each with dimension $M \times m$. The elements of ...
8
votes
2answers
191 views

generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix the maps $$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto ...
9
votes
1answer
262 views

The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, ...
6
votes
1answer
152 views

Are the integer matrices in SO(3,2) “boundedly generated”?

Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$. (The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...
9
votes
3answers
326 views

Integer matrix that does not belong to a free group of rank 2

I'm given two matrices in $SL_2(\mathbb{Z})$ $$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), \ \ B = \left(\begin{array}{cc} 5 & 3\\ 3 ...
7
votes
1answer
116 views

Efficient SVD of a matrix without some of the columns

I have a matrix $A \in \mathbb{R}^{p \times q}$ of rank $r$ and its SVD decomposition, i.e, $$ A = U S V^\top, $$ where $U \in \mathbb{R}^{p \times r}$ and $V \in \mathbb{R}^{q \times r}$ are ...
1
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0answers
108 views

Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$

The problem may be formulated as follows: We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...
17
votes
1answer
453 views

Almost commuting unitary matrices

Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
-1
votes
1answer
43 views

finding a unitary submatrix inside a random matrix

Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...
2
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0answers
28 views

Nonconvex function on the singular value vectors of projected matrix

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...
-1
votes
1answer
185 views

How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]

(Sorry, I do hate editing this many many times but let me try the last time) Gelfand's formula says that $$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$ I am wondering whether there is any ...
11
votes
1answer
405 views

Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$ Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds. Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
1
vote
0answers
138 views

Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$ I know that the sub-representations of a reducible representation ...
4
votes
2answers
198 views

Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion ...
5
votes
1answer
112 views

What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix ...
9
votes
2answers
353 views

Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?

When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...
7
votes
1answer
272 views

Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M det(A)$ and the determinant of its ...
11
votes
1answer
233 views

Have “sturdy squares” been studied before?

Over on PPCG I've just made a question that involves arranging the numbers 1 to 9 on a 3 by 3 grid such that every 2 by 2 subgrid has the same sum. I'm calling such 3 by 3 grids and their N by N ...
7
votes
0answers
120 views

A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$

In the appendix of the paper by Tolhuizen ( http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...
1
vote
0answers
100 views

For of a special case of $Ax>=b$, are there always integer solutions?

The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$. The output is a matrix $A(n\times n)$ ...
1
vote
0answers
46 views

Product of elementary divisors

Let $A$ be an $(m \times n)$ integer matrix (if it helps, we can assume that a is a square matrix). Let $d_i,\ldots,d_s$ be the elementary divisors of $A$. I am interested in the product ...
1
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0answers
55 views

Log convexity for the norm of a vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory. Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
4
votes
0answers
71 views

How does scaling rows to sum to 1, of a positive matrix change the perron vector?

Reposting from math.sx due to lack of response. Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron ...
5
votes
0answers
371 views

How the idea of adjugate matrix has been designed? [closed]

I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...
2
votes
0answers
53 views

Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
7
votes
0answers
174 views

A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that: For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index ...
9
votes
1answer
132 views

a generalization of gamma matrices

Is it possible to find matrix solutions to the following : $$\left(\sum_1^m M_k x_k\right)^n=\left(\sum_1^m x_k^n\right)I_d$$ where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) ...
7
votes
0answers
401 views

Name for an operation on matrices?

Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with ...
2
votes
1answer
79 views

The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime? Let $p_1, \dots, p_r$ be $r$ distinct odd primes. Set $$G = ...
7
votes
1answer
111 views

What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest ...
2
votes
0answers
53 views

Integral of a parametrized commutator

I am trying to solve the following integral $$ \int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2} $$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, ...
6
votes
1answer
121 views

“Additive version” of Kronecker product

Let $A$ and $B$ be two square matrices with complex entries. Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and $\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$. Then the Eigenvalues ...
5
votes
1answer
206 views

Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements. Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ? I am interested in ...
22
votes
3answers
918 views

On determinants formed by binomial coefficients

Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers. Let us form naively a $q \times q$ ...
15
votes
1answer
530 views

Is SL(n,Z[x]) generated by transvections?

Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?
3
votes
0answers
101 views

Relating Numerical Range and Perron-frobenius theorem for positive matrices?

Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers $$W(A)=\{x^HAx\lvert ...
1
vote
1answer
85 views

Is this matrix positive semi-definite? [closed]

Consider $K$ vectors $x_1,\dots,x_K$ in $\mathbb{R}^N$. Define the $K\times K$ matrix $A$ whose $(i,j)$ entry is given as $$A_{ij}=\exp(-\frac{||x_i-x_j||^2}{2})$$ Is this matrix Positive ...
1
vote
0answers
88 views

On triangular Toeplitz matrices

Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist ...
7
votes
1answer
82 views

Approximation theoretic question about operator norm

Let $\|M\|:=\sup_{u:\|u\|=1}\|Mu\|$ be the operator norm induced by the Euclidean distance. Suppose $A$ is a $k\times k$ symmetric matrix with $A_{ij}>0$ for all $i,j$ and $\sum_{i,j} A_{ij} = ...
6
votes
0answers
142 views

Finding $U,V$ in Thompson's Formula

Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that: $e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$ Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x ...
16
votes
0answers
529 views

How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
2
votes
0answers
47 views

Find Moment condition for generalized method of moments

Consider a scalar system with 2K outputs and K+2 unknowns $y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$. The variables $n_{k,\ell}$ are zero mean noise variables. To estimate $a_1$ and $a_2$, ...
7
votes
0answers
123 views

Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...
1
vote
1answer
160 views

characterize certain type of matrices

I am trying to characterize matrices with a certain property : Define $U$ as an $n \times n$ matrix (over C or R; you can also assume that it is unitary or orthogonal if it helps). Now take $n$ ...
-3
votes
1answer
163 views

Eigenvalues of real symmetric matrix [closed]

Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of $A < 2 (n - 1).$
3
votes
1answer
171 views

Is there a bound on the rank of finite index subgroup of SL_3(Z)?

Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?