All Questions
6,026 questions
2
votes
0
answers
104
views
Find an $a \times m$ submatrix of an $n \times m$ matrix with smallest rank
Given a matrix $n \times m$, I want to find the submatrices $a \times m$ by selecting $a$ columns such that their rank is minimal. Can this problem be solved efficiently?
- 20.2k
3
votes
0
answers
57
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 410
25
votes
4
answers
7k
views
"Natural" pairings between exterior powers of a vector space and its dual
Let $V$ be a finite-dimensional vector space over a field $k$, $v_1, \dotsc v_n \in V$ a set of vectors, and $f_1, \dotsc f_n \in V^{\ast}$ a set of covectors. Up to permutation, there seem to be at ...
- 572
0
votes
2
answers
97
views
Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices
I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.
Here's the setup:
I have a graph $G$ represented by a $D\...
- 20.2k
0
votes
0
answers
44
views
Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
0
votes
1
answer
98
views
Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 13.2k
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
- 12.9k
2
votes
1
answer
117
views
When is a $p$-th order stationary point of a polynomial actually a local minimum?
Definition: For integer $p\geq 1$, we say $x\in \mathbb{R}^d$ is a $p$-th order stationary point of a function $f \colon \mathbb{R}^d \to \mathbb{R}$ if there exists a $C>0$ and an $\epsilon>0$ ...
CommunityBot
- 1
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
5
votes
1
answer
539
views
Under what circumstances Is a symmetric matrix representable as a Coulomb matrix?
Question:
I am exploring a neural network architecture inspired by physical interactions, where each neuron has associated "mass" and "position" vectors. The weight matrix between ...
- 20.2k
2
votes
0
answers
89
views
The linear independence and linear elimination of non-crossing matching polynomials
Consider the polynomial set:
$$
f_{ij} = (t_i - t_j)x_i x_j + x_i - x_j, \quad (1 \leq j < i \leq 2n)
$$
where $ t_1, t_2, \dots, t_{2n} $ are pairwise distinct.
Let's look at the non-crossing ...
- 21
1
vote
2
answers
184
views
Generate a low-rank sparse covariance matrix
May I ask how to generate a low-rank sparse covariance matrix? Thank you!
CommunityBot
- 1
8
votes
1
answer
882
views
Is there a conceptual reason why every square complex matrix is similar to a complex-symmetric matrix?
The question is maybe a bit vague, but like the title says: Every square complex-matrix $M$ is equal to $P S P^{-1}$ where $S = S^T$. The proof begins by taking the Jordan Normal Form of $M$, and then ...
- 156k
1
vote
0
answers
93
views
Similarity of non-standard matrices
I am researching numerical methods for PDEs. I particular, I am looking at methods for the linear hyperbolic PDE
$$
u_t+au_x=0.
$$
This is a common approach, because successful methods for this model ...
- 111
7
votes
1
answer
292
views
Existence of matrix diagonalizing $x A + y B$ for all $x, y$ and independent of $x, y$
Let $A_1, A_2$ and $B_1, B_2$ be real symmetric matrices.
Suppose $x A_1 + y B_1$ is cospectral with $x A_2 + y B_2$ for all real numbers $x, y$.
Is it true that there exists a fixed orthgonoal matrix ...
- 1,538
0
votes
1
answer
140
views
Finding positive vectors of a special LGS
Let the following $4 \times 4$ LGS be given for which all coefficients
$a_1, a_2, a_3, a_{11}, a_{12}, ..., a_{33}$ are $>0$:
$a_1 + a_{11} \; x_1 + a_{12} \; x_2 + a_{13} \; x_3 + 0 \; x_4 = (a_{...
CommunityBot
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1
vote
1
answer
91
views
Positive definite kernels on compact interval $[0,1]$
From How to prove that a kernel is positive definite? I learned that a function $f:[0,\infty)\to\mathbb{R}$ induces a positive definite kernel $K:\mathbb{R}^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ ...
- 216
2
votes
1
answer
210
views
Maximum number of ones in a full rank matrix with a restriction
Consider $n \times n$ binary matrices. I am interested in the largest number of ones possible in an $n \times n$ binary matrix with full rank over the field of integers mod 2 with the following ...
- 23.7k
1
vote
0
answers
40
views
Asymptotic unitary invariance of rank-one spiked Gaussian matrix
I'm working on some Random Matrix Theory related stuff for my thesis, and i've come across the following problem:
Consider a (normalized) spiked Wigner matrix $\mathbf{A}$
$$ \mathbf{A} = \frac{\beta}{...
- 63.9k
4
votes
1
answer
236
views
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 12.9k
0
votes
0
answers
50
views
Eigenvalues of functions on finite discrete sets
Suppose I have an arbitrary function on a finite and discrete set $S$ defined as
$$f: S \times S \to \mathbb{C}^{|S|\times |S|}.$$
The $|S| \times |S|$ matrix $M$ is defined as
$$(M)_{ij}=f(s_i, s_j) \...
- 1
2
votes
1
answer
200
views
Upper bound for the rank of a Gram-type matrix
Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
CommunityBot
- 1
10
votes
2
answers
635
views
Largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$?
What is known about the largest set of $k$-wise linearly independent vectors in $\mathbb F_q^n$? I am especially interested when $q=2$, and in the regime where $k$ is fixed an $n\to\infty$. Here are ...
- 46
4
votes
2
answers
587
views
Is this function injective?
For all given ordered lists $$\mathcal A=\big\{\{a_\mu\mid\mu=1,\cdots,N\}\mid\forall\mu,\nu> \mu,\ a_\mu > a_\nu\big\},$$ the function on the quotient space $$ G_\mu(a+\mathbb R: \mathcal A / \...
- 6,696
2
votes
1
answer
547
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
CommunityBot
- 1
3
votes
0
answers
181
views
Levelled trees and the homotopy transfer theorem
In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
- 215
2
votes
0
answers
67
views
Preserving invertibility with adding rows
Suppose I have two $m\times n$ matrices $A$ and $B$ such that an $m\times m$ submatrix of $A$ is invertible if and only if the corresponding $m \times m$ submatrix of $B$ is. Now let's say I append a ...
- 21
0
votes
1
answer
91
views
Finite projective geometry and the Krasner hyperfield
The Krasner hyperfield is an algebraic structure of two operations on $K=\{0,1\}$ called $+\colon K\times K\to \mathcal{P}(K)$ and $\cdot\colon K\times K\to K$ with
$0+0=0$
$0+1=1+0=1$
$1+1=\{0,1\}$
...
- 36
1
vote
0
answers
76
views
What is the operator norm of the sedenions and beyond?
Suppose that $K$ is a field. Then for all $n$, define a bilinear operation $*$ (or $*_{n,K}$ in case there may be ambiguity) on $K^{2^n}$ along with a conjugation operation $^*$ on $K^{2^n}$ by ...
0
votes
0
answers
60
views
The generalized Laplace expansion for tensor
I'm reading this paper https://arxiv.org/abs/1308.3860.
In the Appendix (page 22), the author uses a generalized Laplace expansion for the determinant tensor, as shown in the picture1.
But I only ...
- 1
2
votes
1
answer
247
views
Linear system with matrix as a variable
I have the following two linear systems:
$$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$
$$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$
Both $A,B$ are $2 \times 2$ matrices ...
CommunityBot
- 1
1
vote
1
answer
185
views
A system of linear equations with way too many unknowns — constructing a bivariate distribution from marginals and "the diagonal"
Suppose we are given information about distributions of random permutations $\sigma, \tau : \Omega \to S_n$ as follows:
$$p^1_{k,l} = \mathbb P(\sigma(k) = l), p^2_{k',l'} = \mathbb P(\tau(k) = l), p^{...
CommunityBot
- 1
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 321
4
votes
1
answer
228
views
A question on eigenvalue of parametric matrix
Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
CommunityBot
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0
votes
1
answer
310
views
Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
CommunityBot
- 1
2
votes
1
answer
358
views
q-polynomials in terms of a basis
Consider the polynomials
$$f_n(q)=\prod_{j=1}^n(1+q^j) \qquad \text{and} \qquad g_m(q)=1+q+q^2+\cdots+q^m.$$
I'll list a few examples to motivate my question. Direct calculations show that
$$f_1=g_1, \...
CommunityBot
- 1
15
votes
1
answer
518
views
Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 38.6k
0
votes
0
answers
51
views
Degree of determinant of a (non-monic) matrix polynomial
Let $n=2, 3, \dots$ and consider the matrix polynomial $L(\lambda)=\sum_{k=0}^{\ell}A_k\lambda^k$, where $A_k \in \mathbb{C}^{n\times n}$.
In the so-called monic case (or that can be made monic by ...
- 11
1
vote
1
answer
310
views
Trees and spans of edge labels
Let $T$ be a rooted tree with $m$ leaves. Label every edge with a label of the form $x_i$ or $-x_i$, for some letter $x_i$. For each leaf in the tree, consider the formal linear combination $v$ ...
CommunityBot
- 1
0
votes
0
answers
28
views
Constructing random graphs with given eigenvalues and eigenvectors
In Linial's presentation on SOME PROBLEMS AND RESULTS IN THE
GEOMETRY OF GRAPHS, on slide 7, some relations of properties of graphs to the eigenvalues of their adjacency matrix are listed, e.g.
if $G$...
- 13.2k
3
votes
2
answers
257
views
On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$
I have made the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...
1
vote
0
answers
100
views
PageRank in directed graphs: equivalence of iterative and eigenvalue methods
Given a directed graph $ G $ with $ n $ nodes, we can represent this graph using an adjacency matrix $ A $. The stochastic matrix $ S $ can be derived from the adjacency matrix using the following ...
- 4,058
23
votes
2
answers
3k
views
Formula expressing symmetric polynomials of eigenvalues as sum of determinants
The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
- 188k
1
vote
0
answers
204
views
The wedge product of two positive forms is positive
I have previously posted this question on MSE, but still didn't solve it.
Definition. A real $(p, p)$-form $\psi$ on a complex manifold $M^{n}$ is said to be (semi-) positive, if for any $x \in M$, ...
- 195
2
votes
2
answers
2k
views
How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?
$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists?
I am interested in this question, because my ...
CommunityBot
- 1
0
votes
0
answers
15
views
Change in two spectral deviations due to edge deletion in a signed graph
Prove (or disprove) the following. Let $\Sigma=(G,\sigma)$ be a given signed graph. If $\lambda_1\ge\lambda_2\ge\cdots\ge \lambda_n$ and $\mu_1\ge\mu_2\ge\cdots \ge \mu_n$ are the eigenvalues of the ...
- 61
2
votes
1
answer
456
views
Integrality certification for product of two matrices $A B^{-1}$
Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
CommunityBot
- 1
0
votes
0
answers
46
views
What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342
0
votes
1
answer
102
views
Minimally change matrix with determinant 0
In the following matrix equation, all coefficients $a_{ij}>0$ and all $a_i>0$ and the column sums in the matrix $A$ are all 0
(e.g. $-a_{11}+a_{21}+a_{31}=0$, etc.).
This means that
the ...
- 12.9k
3
votes
1
answer
428
views
Minimum upper bound for sum of the entries of the inverse covariance matrix
Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$
and let $\mathbf{K}$ be the following $n \times n$ covariance matrix
$$\mathbf{K} = \...
CommunityBot
- 1