All Questions
6,260 questions
3
votes
1
answer
150
views
Spectrum of the adjacency matrix of certain directed graphs
For an undirected graph $G$, its adjacency matrix $A_G$ is symmetric, and by the (consequence of) spectral theorem, each of its Jordan blocks has size $1$. This is not true for a general directed ...
- 161
20
votes
7
answers
5k
views
Why do infinite-dimensional vector spaces usually have additional structure?
On Mathematics Stack Exchange, I asked the following question: Why are infinite-dimensional vector spaces usually equipped with additional structure? Although it received one good answer, I feel that ...
- 961
1
vote
1
answer
253
views
Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 823
1
vote
0
answers
24
views
One-step vectorial recurrence into multi-step scalar recurrence on commutative rings with boundary conditions
Introduction over unbounded domain
Consider the forward time shift $\mathsf{z}$ acting on a discrete function of time (sequence) $f=(f^n)_{n\in\mathbb{N}}$ as $(\mathsf{z} f)^{n} = f^{n+1}$. Also ...
- 11
1
vote
0
answers
44
views
Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices
Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties:
$\log(L)$ is plurisubharmonic.
$L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630
1
vote
0
answers
155
views
Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
- 461
1
vote
1
answer
147
views
Is $A^2 + (A^2)^t$ Positive Semidefinite? [closed]
Given a matrix $A$ of size $n\times n$, which need not be symmetric, $A^2 + (A^2)^t$ (where $t$ denotes transpose) is certainly symmetric. Is it positive semidefinite?
- 19
1
vote
1
answer
142
views
Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
- 1,207
3
votes
0
answers
453
views
Dimension of a subspace of $n\times n$ real symmetric matrices
Let $n\in \mathbb N.$ Let $W$ be a non-trivial subspace of $n\times n$ symmetric matrices such that for every $x\in \mathbb R^n\setminus \{0\}$ there exists $a_x\in \mathbb R^n\setminus \{0\}$ such ...
- 49
2
votes
0
answers
242
views
matrices with all maximal minors non-singular
I encountered a sequence of matrices whose maximal minors seem to be all non-singular.
In other words, suppose that we have an $m\times n$ matrix with $m \leq n$, then any choice of $m$ columns is ...
- 5,822
4
votes
1
answer
385
views
Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1?
I'm making a research on Galois theory, and found something interesting regarding the ring of symmetric polynomials:
At least up to 5 variables, we can rewrite the elementary symmetric polynomials ...
9
votes
2
answers
959
views
Direct proof of unique invariant distribution for ergodic, positive-recurrent Markov chain
I posted the following question on MSE, feeling that it perhaps isn't research level mathematics, but didn't get any bites. So, I am crossposting here.
The following ergodic theorem is well known.
...
- 165
6
votes
2
answers
254
views
Eigenvalues of polynomials of two matrices
In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting.
Let $\chi(M)$ denote the characteristic ...
- 37.7k
0
votes
0
answers
99
views
Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
2
votes
0
answers
270
views
Singular values of Kronecker product of random matrices
I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
- 121
4
votes
1
answer
183
views
What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?
Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
- 143
0
votes
0
answers
92
views
Finding a point that minimizes sum of distances to a given set of lines
Given a set $L$ of size $n$ of lines in $\mathbb{R}^d$, find a point $x \in \mathbb{R}^d$ that minimizes: $$\sum\limits_{l\in L}\min\limits_{y\in l} {\lvert \lvert x-y \rvert\rvert}^2$$
I wrote a 1.5-...
- 19
6
votes
2
answers
644
views
Sprinkling signs in unitary matrices
Let $A$, $B$ be $n\times n$ unitary complex matrices, such that for all indices $i,j$ we have $|a_{ij}|=|b_{ij}|$. Does there then exist diagonal unitary matrices $D,D’$ such that $DAD’=B$?
This can ...
- 1,949
4
votes
0
answers
148
views
Frobenius norm bounds on exponentials of anti-Hermitian matrices
Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
- 41
8
votes
1
answer
884
views
Is there a conceptual reason why every square complex matrix is similar to a complex-symmetric matrix?
The question is maybe a bit vague, but like the title says: Every square complex-matrix $M$ is equal to $P S P^{-1}$ where $S = S^T$. The proof begins by taking the Jordan Normal Form of $M$, and then ...
- 4,943
7
votes
0
answers
220
views
Why are these two determinants equal?
This question is a follow up on Mark Wildon's comment from an earlier MO question.
As usual, let $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ with $(q)_0:=1$. Also, define the Gaussian polynomials by
$$\binom{n}...
- 43.2k
2
votes
1
answer
132
views
Name for a sum of dyadic vector products
Question:
is there a name for the following operation
$$\sum_{i=1}^n\sum_{j=1}^mx_iy_j^T,\ x_i,y_j\in \mathbb{R}^k$$ i.e. for generating a square matrix that is the sum of the cartesian product of a ...
- 13.2k
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
2
votes
2
answers
215
views
How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
- 287
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
5
votes
0
answers
181
views
Is the matrix multiplication exponent $\omega$ independent from the choice of the base field
The matrix multiplication exponent, usually denoted by $\omega_{F}$, is the smallest real number for which any two $n\times n$ matrices over a field $F$ can be multiplied together using ${\...
- 151
2
votes
1
answer
123
views
Polar decomposition with respect to the nonstandard involution of quaternionic matrices?
The quaternions admit infinitely many involutions. But up to isomorphism, there are only two: The standard one $t+xi+yj+zk\mapsto t-xi-yj-zk$ and the nonstandard one $\phi:t+xi+yj+zk\mapsto t-xi+yj+zk$...
- 4,943
4
votes
1
answer
250
views
Does a subset with small cardinality represent the whole set?
Assume that we have heavy-tailed distribution $F(x)$ such that
\begin{align}
F(x)=\mathbb{P}[X\geq x]=x^{-0.5}.
\end{align}
Then, we produce $N$ independent samples $X_1,X_2,\ldots,X_N$ from this ...
- 287
2
votes
1
answer
131
views
Is an almost-solvable linear equation with integer coefficients solvable?
Let $M$ be a fixed $m \times n$ rectangular matrix ($m > n$) with non-negative integer coefficients.
Does there exist a pair $(R, \epsilon)$ with the following properties:
If $b$ is a $m \times 1$ ...
- 111
4
votes
1
answer
263
views
Geodesics on orthogonal matrix
Let $ O(n) $ be the manifold of orthornormal matrix, i.e.
$$
O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}.
$$
Then $ O(n) $ is a submanifold of $ \mathbb{R}^{n\times n} $. On $ O(n) $, there is a ...
- 1,219
0
votes
0
answers
87
views
Number of solution to homogeneous linear Diophantine equations
Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form
$a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $
with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
- 11
7
votes
1
answer
331
views
Monomials of linearly independent polynomials remains independent?
I met with a very elementary question on linear algebra which turns out to be quite resistant when I work on it. Any comments or references on either solutions or the possible value of the question ...
- 357
0
votes
1
answer
232
views
Norm equivalence in finite dimensions - is the equivalence "universal" if the dimension is fixed?
I am aware that in a finite dimensional vector space, any two norms are equivalent.
However, I cannot really figure out how "universal" the equivalence constants are.
To be specific, let us ...
- 3,507
2
votes
2
answers
190
views
Do positive-definite elements in finite-dimensional $*$-algebras over $\mathbb R$ always admit square roots?
Let $A$ be a finite-dimensional $*$-algebra over $\mathbb R$. We say that an element $x \in A$ is positive definite if $x$ admits an inverse and if $x = y y^*$ for some $y \in A$. Does every such $x$ ...
- 4,943
5
votes
1
answer
141
views
On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
- 716
7
votes
2
answers
406
views
Decomposition of tensors into symmetry classes according to Schur functors
I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree.
As it is well-known and extremely easy to ...
- 3,307
5
votes
1
answer
136
views
Estimating a symmetric positive-definite matrix from list of matrix vector products
I have a symmetric positive definite matrix (hessian) $H$ which is unknown and expensive to compute explicitly (circa 30*30)
Indirectly in my code I have a growing list of pairs of unit vectors $u_i$, ...
- 71
8
votes
1
answer
325
views
On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
- 128k
5
votes
1
answer
510
views
A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
1
vote
0
answers
152
views
Constrained trace optimization with relavance to optimal asset selection
Let $D$ and $Q$ be two real $m\times m$ diagonal matrices given
$$
D=\left(\begin{array}{cccc}
d_1 & 0 & \cdots & 0\\
0 & d_2 & \cdots & 0\\
\vdots & \vdots & \ddots &...
- 43
1
vote
0
answers
64
views
Maximize trace of precision matrix
Let Q be the uniform distribution over the hypercube $\{1, -1\}^{d}$.
Let P be any distribution that has support including the hypercube.
Define $\Sigma=\mathbb{E}_{x \sim P}[x x^\top]$.
We'd like to ...
- 515
1
vote
0
answers
51
views
Closed form of the product of these $2\times 2$ matrices
I have a series of $2\times 2$ matrices denoted by
$$
M_j=\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}+a_j\begin{pmatrix}-e^{i\theta} & e^{i\theta} \\ e^{-i\theta} & ...
- 375
1
vote
0
answers
109
views
Problems Correction of "Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "' [closed]
Where I can find the problems correction of this book " Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "
- 11
2
votes
0
answers
258
views
Orthogonal complement of arbitrary intersection of Hilbert subspaces
Let $H$ a Hilbert space, and $\mathcal C$ an arbitrary set of closed subspaces of $H$. Is it true that
$$\left( \bigcap_{Z\in \mathcal C}Z\right)^\perp = \overline{\sum_{Z\in \mathcal C} Z^\perp}$$
...
- 63
1
vote
0
answers
329
views
The geometrical multiplicity of the nilpotent matrices
The following point is well-known in the literature.
Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
- 4,058
1
vote
0
answers
158
views
Hankel transform of certain $\pm1$ sequences
The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$
where $s_2(k)$ is ...
- 43.2k
6
votes
1
answer
519
views
Cauchy-Schwarz-like inequality with a power $p$ term
We set :
$\phi_1, \phi_2 : \mathbb{R} \to \mathbb{R}^M$ with compact support
$\theta \in \mathbb{R}^M$ non-zero, $\theta_{p-1} = (\operatorname{sign}(\theta_i)|\theta_i|^{p-1})_{1 \leq i \leq M} \in \...
- 121
5
votes
2
answers
318
views
An inequality problem for certain positive-definite matrices
Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $<0$. Let $a$ be a column $n\times1$ matrix such ...
- 128k