All Questions
Tagged with matrices linear-algebra
1,683 questions
2
votes
1
answer
336
views
Bringing a (Least Squares Problem) Matrix into Block Upper-triangular Shape via Matrix-reordering
I have the problem of solving very large and very sparse least squares problems and, a bit dissatisfied with the run-times of the full-fledged QR-algorithm, I would like to bring the instances into ...
- 13.2k
9
votes
2
answers
900
views
Compute adjugate matrix over commutative ring
Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix.
My current approach is to use the Cayley-Hamilton theorem:
$$\text{adj}(A) = -(A^...
6
votes
2
answers
4k
views
Characteristic polynomial of Kronecker/tensor product
This was asked before on stackexchange but no answer was given. The question is the following:
Let $A$ and $B$ be matrices in $GL(n)$ and $GL(m)$ respectively. Their tensor product $A\otimes B$ is ...
- 3,799
6
votes
0
answers
721
views
Sum of the entries of the inverse covariance matrix
Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = \left[sinc\left(\frac{T\left(r-s\right)}{n}\right)\right]^n_{r,s=...
- 159
8
votes
2
answers
362
views
Unitary factor in polar decompositions
Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant $...
- 1,748
3
votes
0
answers
611
views
Can anyone help me deduce a matrix inequality?
The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying $F^{...
- 103
-2
votes
3
answers
447
views
Determinant of matrix from set {-1, 1} [closed]
Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...
- 45
3
votes
1
answer
148
views
Characterizing space that preserves positive-definiteness property
Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...
- 133
1
vote
0
answers
53
views
Distributing partially known data between n parties
Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ elements....
- 323
4
votes
0
answers
164
views
Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup
I tried asking this on math exchange, but no luck, so thought I'd try here.
Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...
- 41
3
votes
0
answers
149
views
Copositivity under tensor products
Is there any symmetric real matrix $A$ such that $A^{\otimes n}$ is copositive for all positive integers $n$, but such that A is neither positive semidefinite nor has just non-negative entries?
...
- 31
3
votes
0
answers
257
views
Commutative decomposition for full-rank $A$ and low-rank $B$ matrices that do not commute
1. Motivation
Consider symmetric matrices $A,B\in\mathbb{R}^{n\times n}$, and let $A$ be full-rank and $B$ be low-rank. The simultaneous block-diagonalization, defined as the following
$$A=V_{1}\...
- 570
12
votes
0
answers
314
views
Ratio of entries of A and log A where A is a triangular matrix
Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$...
- 5,622
10
votes
3
answers
2k
views
Partial inverse of a matrix - or does it have its own name?
In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here ...
- 1,612
46
votes
1
answer
6k
views
Determinant of a determinant
Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m \...
- 23.8k
10
votes
2
answers
733
views
Product $PVPVP$ is elementwise nonnegative?
Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
...
- 143
5
votes
1
answer
1k
views
Anti-bidiagonal matrix with main anti-diagonal {1,2,3,...} and first sub-anti-diagonal {-1,-2,-3,...} has eigenvalues lambda={1,-2,3,-4,...}
Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows
$$
B_6=\begin{bmatrix} & & & & & 6\\
& & & &...
- 570
6
votes
1
answer
954
views
Proving that the kernel of this matrix is of dimension 2
(Edit : see at the bottom of the question for an additional surprising possible hint.)
Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when $...
- 403
3
votes
1
answer
180
views
Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?
The nonnegative matrix
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = \...
2
votes
0
answers
318
views
Is there an efficient way to compute the "complete subset regression"?
Background: Let $X \in \mathbb{R}^{N\times K}$ and $y \in \mathbb{R}^{N\times 1}$ be data for a regression problem. The aim is to find $\beta \in \mathbb{R}^{K\times 1}$ such that $X\beta \approx y$ ...
- 326
1
vote
0
answers
459
views
How to restructure adjacency matrix $A$ from shortest distance matrix $B$ in Network topology inference
An undirected graph with $n$ nodes could be referred to as an adjacency matrix $A$. $A=[a_{ij}]_{n×n}$ with $a_{ij}=a_{ji}=1$ standing for there being an edge between node $i$ and node $j$, and no ...
- 287
4
votes
1
answer
1k
views
Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices
Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some ...
- 41
7
votes
4
answers
1k
views
Generalized Cauchy-Binet sum over a fixed subset of indices
I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
- 13.6k
1
vote
0
answers
108
views
MInors related problem [closed]
A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is $||x||=\sqrt{x_1^2+x_2^2+...
- 11
5
votes
3
answers
1k
views
Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?
I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...
- 153
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
2
votes
0
answers
279
views
Eigenvalues of this matrix
I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$
Let me give a ...
- 329
-1
votes
1
answer
142
views
Action of rotation group on Matrices [closed]
Is the following assertion true?
Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in SO(p,\...
- 601
27
votes
3
answers
13k
views
What is known about the distribution of eigenvectors of random matrices?
Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...
- 433
0
votes
1
answer
1k
views
Symmetric Zero-Diagonal Matrices
Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1_n\\1_n&0\end{array}\right]$. Then $X^t\Omega X$ is ...
- 1,090
3
votes
0
answers
2k
views
Bounds on the effect of a matrix product on the Frobenius norm
I was wondering if there was a way to put upper and lower bounds on the Frobenius norm of a matrix product in relation to the Frobenios norm of one of the individual matrices, i.e,
$c_1\|A\|_F\le\|AS\...
- 31
16
votes
0
answers
808
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
7
votes
3
answers
5k
views
Proof for a rank-one decomposition theorem of positive (semi) definite matrices
$\DeclareMathOperator\rank{rank}\DeclareMathOperator\trace{trace}$Consider the following result which I recently came across in a research paper in my area (signal processing)
Let $X$ be a $N\times N$...
- 1,421
1
vote
1
answer
6k
views
Largest eigenvalue of the sum of Hermitian matrices [closed]
Is there an expression for the largest eigenvalue of the sum of two Hermitian matrices in terms of the spectrum of the same matrices?
- 2,099
9
votes
0
answers
978
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
15
votes
3
answers
3k
views
Determinant of a $k \times k$ block matrix
Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...
- 301
2
votes
1
answer
265
views
How can I prove that the negative biased triangular kernel is positive semidefinite
How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$
$k(x, x') = (1 - 2|x-x'|)$
is a positive semidefinite function?
It turns out to be psd function when ...
5
votes
1
answer
3k
views
Operator norm vs spectral radius for positive matrices
I believe the following statement should be true but somehow I don't see an argument:
For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
- 421
4
votes
1
answer
594
views
Smith Normal Form for block matrices over the integers
Are there any known results on the Smith Normal Form for block matrices over the integers?
In particular, I am interested in matrices of size $kr \times ks$ made of square blocks of size $k$ such that ...
- 41
0
votes
1
answer
154
views
Dimension of a similarity class
Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
- 3,741
2
votes
1
answer
1k
views
An inequality involving traces and matrix inversions
The following question kept me wondering for some time:
Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
- 53
2
votes
1
answer
346
views
Alike looking matrices imply convergence of eigenvalues?
This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...
user54300
3
votes
1
answer
586
views
On the solution of a generalized Lyapunov equation
We shall reconsider the following equation
$$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$
where $p$ is a positive integer and $C$ is a known symmetric positive semidefinite matrix.
I met with this ...
- 77
5
votes
1
answer
611
views
Die hard nilpotent spaces
Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is nilpotent if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned ...
user56203
10
votes
1
answer
836
views
Factor a sum of products of cofactors
Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...
- 2,902
3
votes
0
answers
128
views
Reducing $\ell_1$ norm of non-full-rank matrices
I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ${\...
- 291
10
votes
1
answer
938
views
Why does this antisymmetric product factor out a determinant?
Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...
- 2,902
5
votes
3
answers
693
views
Norm of triangular truncation operator on rank deficient matrices
Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ A\|\le\frac{\...
- 153
0
votes
1
answer
1k
views
How do eigenvalues change if we duplicate a row and column of a symmetric matrix?
Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix,
$$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
3
votes
1
answer
320
views
Number of Matrices with bounded determinant
Here's my question:
Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
- 31