All Questions
6,292 questions
-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
3
votes
3
answers
2k
views
When does a Vandermonde-like matrix have full rank
I have a matrix which is similar to Vandermonde matrix except that the entries are monomials of degree $d$ polynomial in 2 variables. Each row has the following form:
$X_{i}= [1, x_{i}, y_{i}, x_{i}^...
- 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
8
votes
2
answers
5k
views
About Sylvester's determinant
If $A$ is any $n \times m$ matrix and $B$ is any $m \times n$ matrix then one familiar form of the Sylvester's identity is $\det(I + AB) = \det(I + BA)$.
Now somehow curiously this above identity is ...
- 617
5
votes
1
answer
441
views
Dirichlet Characters as Eigenvectors
This was asked in Math Stackexchange here but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at ...
- 10.4k
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
1
vote
0
answers
350
views
Pre- and post-multiplication by diagonal matrices [closed]
Let $\mathbf{1}$ denote an $n\times 1$ vector with all entries equal to 1. Given an $n\times n$ matrix A with strictly positive entries, and non-negative diagonal matrices $D_1$ and $D_2,$ evaluate ...
- 1,269
17
votes
6
answers
3k
views
Does the linear automorphism group determine the vector space?
I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...
- 12.9k
1
vote
1
answer
255
views
Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?
Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...
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
5
votes
1
answer
602
views
minimal polynomial for a graph
I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...
- 51
2
votes
1
answer
211
views
Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?
Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.
Consider a equivalent relation on $R\mathbb{Z}^N$ defined by
$a\sim b$ if $a-b\in \...
- 31
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
52
votes
5
answers
4k
views
When exactly and why did matrix multiplication become a part of the undergraduate curriculum?
The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. A few weeks later, Born and Jordan read this work and recognized matrix ...
- 91.8k
1
vote
0
answers
129
views
Decomposition of non-singular matrix [closed]
Is there any way to show that a non-singular matrix A can be partitioned as follows:
\begin{eqnarray*}
A&=&\left[
\begin{array}{cc}
\underset{\left( k\times k_{1}\right) }{{B}_{11}}\underset{\...
- 11
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
7
votes
1
answer
2k
views
If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?
Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
- 483
5
votes
3
answers
461
views
Minimize distance between centroids of subsets of points
In a n-dimensional space, I want to divide a set of m points into v (non-empty) subsets.
I want to minimize the sum of the pairwise Euclidean distances between the centroids of the resulting subsets.
...
- 51
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
11
votes
2
answers
3k
views
Do singular values dominate eigenvalues?
Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots \...
- 1,135
4
votes
2
answers
316
views
Question on Posets and open sets [closed]
i'm sorry if my question is really trivial but this one is really bugging me out..
So let's have a partially ordered set $I$ with the topology in which the open sets are the increasing ones: $i\in U$ ...
- 609
8
votes
2
answers
246
views
Are sums of 0-1 Pareto efficient vectors Pareto efficient?
Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...
- 388
1
vote
1
answer
231
views
Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?
The notation ${}^t g$ for the transpose of a linear transformation is, in my view, quite unusual: otherwise (at least in many areas of math), one almost never sees subscripts or superscripts appearing ...
- 7,347
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
15
votes
2
answers
6k
views
Linearly constrained eigenvalue problem
Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& \...
- 723
1
vote
0
answers
460
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
1
vote
1
answer
72
views
Looking for algorithms based on sorting [closed]
i am looking for algorithms which use sorting in low-dimensional space like $R$ and how they are generalized for higher-dimensional spaces like $R^2$ where there is no sorting possible. (i.e. numbers ...
- 11
5
votes
2
answers
1k
views
eigenvalue estimate of the adjacency matrix
The adjacency matrix of a nonempty (undirected) graph has a strictly positive largest eigenvalue $\lambda_\max$. A very easy upper estimate for it can be obtained directly by Gershgorin's theorem:
$$
...
- 3,388
21
votes
1
answer
564
views
Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 19.6k
1
vote
1
answer
558
views
Is there a Gröbner basis analogue that exists for vector spaces?
Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$.
...
- 131
1
vote
2
answers
747
views
Can any antilinear involution be trivialized by a change of basis?
Consider an antilinear involution, that is an antilinear map on a complex vector space, whose matrix $M$ obeys $MM^*=1$ where the star denotes complex conjugation. Can we find a change of basis whose ...
- 378
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
2
votes
2
answers
286
views
Linearly independent family of sequences of rationals with a cardinal equal to the continuum
I'm coming back to this question. Is it possible to have "an explicit" linearly independent family of sequences of rationals with a cardinal equal to the continuum?
PS: sorry for the duplicate on the ...
- 1,355
2
votes
0
answers
1k
views
Random matrices whose limit gives exact Wigner surmise
Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write $\rho^W_0(s)=\...
- 4,952
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)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
0
votes
0
answers
153
views
extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 1,835
2
votes
0
answers
279
views
Eigenvalues of this matrix
I have a linear map that is defined by $$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}$$
Let me give a ...
- 329
7
votes
1
answer
197
views
Compute only selected components of an eigenvector
I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...
- 91
5
votes
1
answer
365
views
A specific Vandermond matrix [closed]
Consider the Vandermond matrix
$$
V (x_1, x_2, \ldots , x_n) =
\begin{pmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\
1 &...
user45183
4
votes
1
answer
181
views
Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$
I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
- 287
-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