All Questions
3,208 questions
5
votes
1
answer
103
views
Interpolation between two matrices so that $L^p$ norm is controlled
Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
- 697
0
votes
1
answer
525
views
What is the mathematician's definition of the determinant? [closed]
I am trying really hard to find a good definition of the determinant.
I have looked virtually every single resource online and everybody gives a different answer:
sum of cofactors or minors https://...
- 179
7
votes
1
answer
305
views
Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric
Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
- 173
4
votes
0
answers
262
views
Two questions about three circulant matrices
Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$
$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$
where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
- 696
6
votes
2
answers
647
views
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this ...
- 79
0
votes
0
answers
49
views
Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant
Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
0
votes
0
answers
131
views
On a matrix equation with Kronecker product
Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
- 271
6
votes
0
answers
188
views
Expressing an invertible sparse matrix as a product of few elementary matrices
Let $M$ be an $n \times n$ matrix with integer entries. Suppose that $M$ is invertible (over the integers) and that $M$ has at most $An$ nonzero entries, each of which is less than $B$ in absolute ...
- 18.7k
0
votes
0
answers
85
views
Symmetric indefinite matrix of fixed rank — manifold structure?
I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g....
0
votes
1
answer
104
views
Change of the smallest positive eigenvalue after a rank-one update
Given natural numbers $n,r,R\in\mathbb{N}$ with $r,R\le n$, let $A\in\mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{n\times R}$ be two matrices with full column rank and let $c\in\mathbb{R}^n$.
Denote ...
0
votes
0
answers
33
views
Elliptic operators with Robin boundary conditions
Can it be proved that two elliptic operators with Robin boundary conditions generate an interval $P$-matrix?
$$
-a\Delta u_i = f_i, \quad a\frac{\partial u_i}{\partial n} + bu_i = 0
$$
$$
-a\Delta v_i ...
- 243
2
votes
1
answer
264
views
Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
- 637
3
votes
1
answer
155
views
Does this matrix equation always have a solution?
Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example,
$A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
- 235
1
vote
1
answer
44
views
Wold decomposition of toral endomorphisms
Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^...
1
vote
0
answers
72
views
Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
- 111
3
votes
1
answer
130
views
Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices
Consider a collection of positive definite matrices $\{A_1,...,A_n\}\in\mathbb{S}_{++}^d$, the Karcher mean of these matrices is given by (see (5.4) in [1]):
$$
\min_{X\in\mathbb{S}_{++}^d} f(X):=\...
- 125
1
vote
0
answers
97
views
Non-vanishing principal minors up to swapping columns
An undergraduate student asked me the following seemingly easy question. After a few days of thinking, I still couldn't come up with an answer, nor could I find one online. Maybe folks here could help?...
- 403
9
votes
2
answers
646
views
Are these two methods for constructing Hadamard matrices known?
These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:
Context:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this ...
- 3,054
1
vote
0
answers
81
views
Pre-positive definite functions?
A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
- 620
0
votes
1
answer
134
views
Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
2
votes
1
answer
226
views
Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?
I am trying to prove that the function:
$$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$
is a positive definite function over the natural numbers. What has sometimes ...
- 3,054
8
votes
3
answers
595
views
Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
- 1,459
1
vote
1
answer
270
views
Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$
I have a hard time solving the following two matrix equations for unknown permutation matrix $X \in \mathbb{R}^{n \times n}$:
$$X^T A X = B_1$$
$$X A X^T = B_2$$
where, $A$, $B_1$ and $B_2$ are all $n ...
- 11
36
votes
4
answers
2k
views
Determinant of the random matrix $X^2+Y^2$
$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.
i) When $n=2,3,4$, one ...
- 3,741
1
vote
1
answer
149
views
Is there a necessary and sufficient condition to determine whether a number sequence can serve as the first few moments of a Radon measure?
Given a few positive numbers $(M_1, M_2,\cdots, M_K)$, they are the moments of a measure if
\begin{equation}
M_k = \int d\mu(x) x^k,\quad k = 1,2,\cdots,K.
\end{equation}
This is related to the ...
- 11
2
votes
0
answers
146
views
What are the name and inverse of an interesting integer matrix?
It is practicable to compute the matrix inverses
\begin{align*}
\begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 2 & 2^2 \\
\end{pmatrix}^{-1}
&=\begin{pmatrix}
1 & 0 &...
- 1,091
5
votes
2
answers
350
views
How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
- 109
1
vote
1
answer
217
views
Perturbation of matrices
Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$.
Question. Does there exist a Lebesgue measurable ...
- 4,135
1
vote
1
answer
214
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
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 $N$ ...
- 93
0
votes
1
answer
744
views
A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?
I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:
The Rydberg formula for ...
- 3,054
2
votes
1
answer
298
views
Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
- 557
1
vote
0
answers
134
views
Matrix valued word embeddings for natural language processing
In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
2
votes
0
answers
114
views
What is the quantity $\sqrt{\frac{c^2+d^2}{a^2+b^2}}$ of a matrix with determinant one?
Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one,
$$
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$
I'm currently working on a problem where I obtained a condition on the ...
- 173
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
2
votes
1
answer
198
views
An inequality related to matrix trace
$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
- 21
4
votes
1
answer
230
views
Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D
Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
- 187
1
vote
1
answer
112
views
square matrix depending on complex value: spectral radius continous? [closed]
Let $A(z)$ be a $n\times n$ square matrix depending on the complex value $z$ and $\lambda_z$ is its spectral radius.
Is $\lambda_z$ continous or is it possible that it can jump? Or maybe someone knows ...
- 83
0
votes
0
answers
92
views
Norm of matrix product sum
Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
- 159
4
votes
1
answer
214
views
Diameter of the unimodular group with Gauss moves
$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.
It is well known that ...
- 327
2
votes
0
answers
137
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 823
1
vote
2
answers
137
views
Methods to solve for a matrix whose entries satisfy certain properties
(This question is a repost of a deleted question I asked, because the previous version had several elements missing)
Setting
For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
- 135
1
vote
1
answer
520
views
Optimal transport between two matrices
I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
- 243
1
vote
0
answers
35
views
Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?
I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
- 111
5
votes
2
answers
508
views
Is there a name for this family of matrices?
Let $0<a_1<a_2<\cdots<a_n$ and let $A$ be the symmetric $n\times n$ matrix with
${ij}^\text{th}$ entry $A_{ij}=\min\{a_i,a_j\}$.
For example, if $a_i=i$ for each $i\le n=5$ then
$$A=\begin{...
- 51
1
vote
1
answer
425
views
How to represent infinite matrices in Mathematica?
I asked this question on Mathematica site and the answer was basically "it is impossible". So, some math theory needs to be thrown in besides coding.
Let us represent infinite matrices as ...
- 10.1k
2
votes
0
answers
51
views
What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
- 283
1
vote
1
answer
252
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
9
votes
1
answer
361
views
Conjugates iff conjugates over $\mathrm{GL}_n(\overline{\mathbb{F}_q})$?
Let $G$ be a connected, almost simple linear algebraic group defined over a finite field $\mathbb{F}_q$. Let $g, g'\in G(\mathbb{F}_q)$ be conjugates by an element of $\mathrm{GL}_n(\overline{\mathbb{...
- 20.2k
0
votes
1
answer
79
views
Matrix sparsity pattern from Boolean condition
I would like to determine the sparsity pattern of a matrix $A_m$, where the entry $(k,l)$ of $A_m$ is non-zero if the condition $\operatorname{test}(k,l,m)$ is true:
$\operatorname{test}(k,l,m)$ = &...
