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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
67 views

The state-transition-matrix of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
1
vote
1answer
87 views

Largest element in inverse of a positive definite symmetric matrix [closed]

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
6
votes
1answer
137 views

Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$). Can we prove that the following problems are equivalent: $$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
2
votes
0answers
65 views

SVD when only off-diagonal terms are known

I have a real matrix $A \in \mathbb{R}^{n\times n}$ such that: $A$ is symmetric All the off-diagonal terms are known and positive Has rank $k<n$ Unfortunately I don't know the values of the ...
19
votes
1answer
1k views

A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...
1
vote
1answer
26 views

how to force least square solution matrix to be diagonal [closed]

Let's say I have the following equation $$AX=B$$ where $A$ is a $8\times 3$ matrix (known), $X$ is a $3\times3$ "diagonal" matrix which represents the coefficients (unknown) and $B$ is a $8\...
1
vote
3answers
124 views

Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
3
votes
1answer
97 views

Wiener-Hopf factorization of matrices

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are ...
2
votes
1answer
92 views

Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form $$ |D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)} $$ providing all values in $(\cdot)>0$. ...
1
vote
1answer
76 views
-1
votes
1answer
114 views

Representations of the $3\times 3$ Heisenberg group [closed]

I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations. Following this article given a symplectic bilinear form $\langle, \...
2
votes
0answers
35 views

Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications? For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature. In some of ...
3
votes
0answers
100 views

An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
5
votes
2answers
350 views

Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries

This is a more carefully worded version of this question, here tailored to professional mathematicians. Consider a matrix ${\bf A}\in{\bf M}_{n\times n}({\mathbb R})$ with possibly positive, negative ...
0
votes
1answer
96 views

Efficient computation of matrix exponential of trace zero matrix [closed]

I am looking for identities that may help with numerical computation of the matrix exponential ${\rm exp}(A)$ where ${\rm tr}(A)=0$. I am already aware of general-purpose algorithms for computing the ...
18
votes
1answer
224 views

How many ways can I factor a matrix (over $\mathbb{Z}$)?

Let $A$ be a fixed matrix in $M_2\mathbb{Z}$ with determinant $n \neq 0$. Question 1 How many ways can I write $A = XY$ for $X, Y \in M_2\mathbb{Z}$? The answer to this question is pretty clearly ...
3
votes
0answers
52 views

Bounding the number of information sets in a linear binary code

A pretty well-known theorem regarding linear $(n,k,d)$ codes is that every $n-d+1$ coordinate positions contain an information set, but not all $n-d$ coordinate positions do. This is equivalent to a ...
1
vote
1answer
130 views

Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using, H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\...
2
votes
0answers
49 views

Riemann theta function with asymptotically large Toeplitz Matrix

As a follow up to How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently Suppose that $M$ is a large Toeplitz matrix. With a suitable scaling $K^{-n}$ for some $K$, what will the Riemann ...
1
vote
0answers
111 views

How to rotate a covariance matrix which contains quaternion elements? [closed]

I am implementing a paper which recovers full-3d body pose from images. It represents individual body parts as 7D vectors containing first the absolute 3D location [x y z] and then the unit ...
6
votes
1answer
125 views

Positivity of power of positive PSD matrices

Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries. Suppose now that, on top of having nonnegative entries, ...
5
votes
0answers
81 views

Tensor matricizations and their decompositions

Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
3
votes
1answer
110 views

Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are $L^...
8
votes
1answer
137 views

Exact eigenvalues of a specific tridiagonal matrix

I'm studying the following tri-diagonal matrix $$ X = \begin{pmatrix} 0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\ x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
4
votes
0answers
95 views

How to enumerate a discrete group of matrices by their Frobenius norm?

Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$, and it is finitely generated by some known generators. That is, $G=\langle g_1,\dots,g_n\rangle$. The Frobenius norm of a matrix $m=\...
3
votes
1answer
113 views

An extremal combinatorics problem involving column summation

Given $n\in\Bbb N$, $\alpha>0$, $\beta\in\big[\frac12,1\big]$ denote $\mathcal R_{n,\alpha,\beta}$ as collection of all $2^n\times 2^{n^\alpha}$ $0/1$ matrices with every row summing to strictly ...
1
vote
0answers
52 views

Asymptotic determinant of $2\times 2$ Toeplitz matrix

The problem that I am dealing with is to compute the determinant of a $2\times 2$ Toeplitz matrix[1] (in general I would like to generalize to a more general case, but let's consider the easiest case ...
3
votes
0answers
63 views

How to construct large sets of $m$-dimensional vectors over a finite field such that any $m$ of them are independent?

Let $m\ge 2$ be an integer and $\mathbb{F}$ be a finite field of order $p^k$. I want to construct as many as possible $m$-dimensional vectors $v_1,v_2,\ldots,v_n$ in field $\mathbb{F}$ such that any $...
0
votes
0answers
36 views

For which matrices deciding permutation similarity is polynomial?

Q1 For which matrices deciding permutation similarity is polynomial? It is not easier than graph isomorphism (and very likely is equivalent to it). If necessary, assume the entries are nonnegative ...
0
votes
1answer
62 views

Eigenvalue-related statements [closed]

(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty) How can I prove that the ...
8
votes
2answers
471 views

A log inequality for positive definite trace-one matrices

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...
0
votes
1answer
83 views

Large Tridiagonal Matrix - Eigenvalues

Consider large tridiagonal matrix (where $a$ and $b$ are real numbers): $M = \begin{pmatrix} a^2 & b & 0 & 0 & \cdots \\ b & (a+1)^2 & b & 0 & \cdots & \\ ...
0
votes
1answer
164 views

$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable. $Z$ is eigenvalue of $P(Z )$ if $\...
4
votes
0answers
160 views

Efficiently calculate the trace of the product of two large but symmetric matrices, one of which is an inverse

Sorry about the long title. I need to calculate the trace of $M(M+D)^{-1}$, where $M$ is a dense symmetric matrix, and $D$ is a diagonal matrix. The main issue is the dimension could be large (usually ...
4
votes
1answer
72 views

spectrum of Hadamard matrices

A (±1)-matrix is a matrix whose entries are 1 and −1. An $n \times n$ (±1)-matrix is called an Hadamard matrix if the rows are orthogonal. Equivalently, An $n \times n$ (±1)-matrix $H$ is Hadamard ⇔ $...
3
votes
2answers
106 views

Eigenspace of convex combination of two idempotent matrices

Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix $$H_\mu:=\mu H_1+(1-\mu)H_2.$$ I'm looking for a description of $\...
5
votes
1answer
85 views

Optimization of a function of a positive definite matrix and its inverse

This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved. Suppose I have two real, positive ...
13
votes
2answers
453 views

A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that $\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
1
vote
0answers
42 views

Complex conjugate and unitary complex conjugate

Definition: Let V be complex finite dimensional inner product space Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$ Definition: Anti-unitary Complex ...
3
votes
2answers
271 views

A question on determinant of a matrix polynomial

Let ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$. ${\rm{P(}}\lambda {\rm{) = ...
19
votes
0answers
574 views

Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
0
votes
0answers
30 views

Distribution of zeros and ones over matrix

I have the following problem: Given a matrix with n rows and m columns. Some elements of the matrix are unavailable. For each column, you have a set containing a number of zeros and ones which must ...
2
votes
0answers
131 views

On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration \begin{equation} X_{k+1}=\frac{1}{N}\sum_{i=1}^...
7
votes
1answer
78 views

Add a multiple of $I$ to a matrix to minimize its operator norm

Given $A\in\mathbb{C}^{n\times n}$, what is $s_* = \arg\min \|A-sI\|$? Here $\|A\|$ is the operator norm, $\|A\|=\rho(A^*A)^{1/2}$, and $I$ is the identity. The corresponding problem for the ...
2
votes
0answers
169 views

Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of $\...
1
vote
0answers
78 views

Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
0
votes
1answer
110 views

Derivative of matrix logarithm w.r.t. real parameter

Let $A(t)$ be an invertible square matrix that depends (differentiably) on a real parameter $t$. It is well known that for example $$ \frac{d}{dt} A(t)^{-1}=-A(t)^{-1}\ \dot{A}(t)\ A(t)^{-1} $$ and $$ ...
7
votes
3answers
194 views

Is there a standard name for the following type of linear operator?

Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...
2
votes
1answer
252 views

A presentation for $GL(2,\mathbb{Z}/p^n \mathbb{Z})$

In 'A presentation of $PGL(2,p)$ with three defining relations' by E.F.Robertson and P.D.Williams, we can find a presentation of $PGL(2,p)$: $\langle a,b | a^2 = b^p = (a b^2 a b^r)^2 = (abab^r)^3 = ...
1
vote
0answers
91 views

Roots of matrices in $G_2(Z)$

Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral ...