All Questions
1,907 questions with no upvoted or accepted answers
6
votes
0
answers
218
views
Reconstruct orthogonal from an orthostochastic matrix
Given an $n \times n$ orthostochastic matrix $\mathbf{A}$, i.e., there exists an orthogonal matrix $\mathbf{O}$ with $A_{ij} = O_{ij}^2$ for all $1\leq i,j \leq n$. What is the fastest way to find $\...
- 909
6
votes
0
answers
441
views
Solve this nonlinear matrix equation?
For $M$ and $N$ two invertible square matrices of the same size $n$, consider the equation
$$
\forall i,j, \quad M_{ij}(M^{-1})_{ji} = N_{ij}(N^{-1})_{ji}\ .
$$
Assuming we know $M$, we want to find ...
- 378
6
votes
0
answers
296
views
formalization of coordinate-free linear algebra in a proof assistant
I am aware of projects that formalize linear algebra in existing proof assistants (i.e. Coq), but it seems like most of them are based on matrices. I was wondering if it's done in a coordinate-free ...
- 161
6
votes
0
answers
106
views
A conjectured trace inequality for some products of powers of matrices II
Let $B, R\in M_{n}(\mathbb{C})$ hermitian and $B$ positive semidefinite.
Let $s \in \mathbb{R}$ and $s \ge 0$ .
Does then hold $Tr[B^s (I + B R^2 B)^{-1}] \le Tr[B^s (I + R B^2 R)^{-1}]$ ?
See also ...
- 2,753
6
votes
0
answers
289
views
Upper bound for $\|\textbf{D}^{-1}\|$, where $\textbf{D}$ is a matrix with specific sparse pattern
Consider the block matrix given by
$$\textbf{D} = \left[
\begin{array}{ccc}
\left[
\begin{array}{ccc}
D & \ldots & D\\
\vdots & \ddots & \vdots\\
D &...
- 143
6
votes
0
answers
133
views
Is the map taking a matrix to its semisimple part algebraic (or at least holomorphic)?
Let $\text{Mat}_n(\mathbb{C})$ be the set of $n \times n$ complex matrices. Let $\sigma\colon \text{Mat}_n(\mathbb{C}) \rightarrow \text{Mat}_n(\mathbb{C})$ be the map that takes a matrix to its ...
- 61
6
votes
0
answers
179
views
Growth in a vector space
I am interested in "growth" in finite vector spaces. I have been wondering about the veracity of the following statement:
Statement: For every $\varepsilon>0$ there exists $b\in\mathbb{Z}^+$ ...
- 11.2k
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
6
votes
0
answers
450
views
Can this nonlinear vector equation be solved analytically?
I have the following vector equation:
$$
{\bf Ax} + {\bf b} + {\bf Cx}^{ \circ - 1} = {\bf 0}_n
$$
Where ${\bf x}$ is a vector of unknown variables. ${\bf b},{\bf x}, {\bf x}^{\circ - 1}, {\bf 0}_n ...
- 235
6
votes
0
answers
215
views
Matrix semigroups in which a weighted average of eigenvalues is multiplicative
A problem in fractal geometry requires me to consider matrix semigroups with the following curious property. For a $d \times d$ real matrix $A$ let $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \...
- 6,206
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 907
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
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
6
votes
0
answers
489
views
Symmetric matrices with $\rho(A)\gg\|A\|_\infty$
For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
- 23k
6
votes
0
answers
375
views
Kasteleyn, Gessel-Viennot and eigenvalues
The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...
- 1,343
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
6
votes
0
answers
514
views
concentration for eigenvectors
I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
- 6,962
6
votes
0
answers
465
views
Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
6
votes
0
answers
267
views
Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
5
votes
0
answers
231
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 309
5
votes
0
answers
582
views
Dimension inequality for subspaces in field extensions
Let $K\subset L$ be a field extension and $A, B\subset L$ be $K$-subspaces of $L$ of finite positive dimensions. Assume further that for every $a, b \in L$ and every nontrivial proper finite ...
- 429
5
votes
0
answers
137
views
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
- 351
5
votes
0
answers
169
views
Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
- 51
5
votes
0
answers
184
views
What is the fastest algorithm for multiplying one given number with many others?
When multiplying two numbers with each other, which are $n$-bit numbers, there are several algorithms like the one of Karatsuba ($O(n^{\log_2 3})$) and a new one doing it even better (Harvey - Van der ...
- 749
5
votes
0
answers
138
views
Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)
Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
- 32.5k
5
votes
0
answers
202
views
Difficulty of solving $Ax=b$ in terms of limiting spectral density of $A$?
Suppose $A$ is a random real-valued $n\times n$ matrix and we want to know the difficulty of solving $Ax=b$ when entries of $b$ are sampled IID from Normal$(0,1)$.
Can we say anything about the ...
- 2,442
5
votes
0
answers
180
views
Is the matrix multiplication exponent $\omega$ independent from the choice of the base field
The matrix multiplication exponent, usually denoted by $\omega_{F}$, is the smallest real number for which any two $n\times n$ matrices over a field $F$ can be multiplied together using ${\...
- 151
5
votes
0
answers
190
views
Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
5
votes
0
answers
171
views
Maximal minors of tensor product
Let $r \leq n$ be integers, and let $A$ be an $r \times n$ integer-valued matrix such that each $r\times r$ minor of $A$ is in $\{0, 1,-1\}$. Is it true that each $r^2 \times r^2$ minor of $A\otimes A$...
- 980
5
votes
0
answers
196
views
Is it true that the $\mathbb{F}_p$-rank of a linear combination of matrices is usually not smaller than its $\mathbb{Q}$-rank?
Consider fixed $3 \times 3$ integer matrices $A_1,A_2,A_3$ and the $\sim H^3$ linear combination matrices $A(\mathbf{h})=h_1A_1+h_2A_2+h_3A_3$ where $h_1,h_2,h_3$ are integers with $\vert h_i\vert \le ...
5
votes
0
answers
447
views
Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form
$H_n(a_0(n))=\begin{pmatrix}
a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\
(1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\
(2!)^2 &...
- 51
5
votes
0
answers
208
views
Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
5
votes
0
answers
835
views
Gershgorin's 2nd theorem (disjoint circles): elementary proof?
Let $A \in \mathbb{C}^{n\times n}$ be a complex matrix. We let $a_{i,j}$ be the $\left(i,j\right)$-th entry of $A$ for all $i, j \in \left[n\right]$ (where $\left[n\right]$ denotes $\left\{1,2,\ldots,...
- 33.8k
5
votes
0
answers
340
views
Does this subset of $\mathbb{Z}$ form a subgroup?
Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ with basis $e_1,\ldots,e_n$. Consider the abelian group $N = \mathbb{Z}e_1\oplus\ldots\oplus\mathbb{Z}e_n$.
Assume that $A$ is any ...
- 329
5
votes
0
answers
241
views
Growth rate of cohomology
Fix a finite dimensional graded vector space $V$, a differential $d$ on $V \otimes V $ $(i.e.\,\, d^2=0\,\, and \,\, deg\, d =1)$ such that $(d_{1,2} \otimes id_3)$ anti-commutes with $((-1)^{deg}_1 \...
- 189
5
votes
0
answers
131
views
What is known about this conjectured symmetry in the generalized Radon-Hurwitz numbers?
The generalized Radon-Hurwitz number $\rho(m, n)$ is defined as the
maximal dimension of a subspace contained in $Q_{m,n }$, the subset of all real $m\times n$ matrices $A$ which satisfy $AA^T=\lambda ...
- 13.4k
5
votes
0
answers
237
views
Linearly independent quadratic forms vanishing on a finite set of points
The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, \cdots, Q_m \in \mathbb{C}[x_0, \cdots,...
- 26.9k
5
votes
0
answers
128
views
If power of two matrices becomes equal then stays equal, with left-side row multiplication
During my research on probabilistic automata (in joint Computer Science and Mathematics), I could reduce a certain problem to a problem of matrices.
We are given the (element-wise) non-negative ...
- 51
5
votes
0
answers
97
views
Periodics of Coxeter matrices for truncated Nakayama algebras
For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$).
So for example for $n=7$ and $r=3$ we ...
- 26.5k
5
votes
0
answers
144
views
Do products of distance functions separate points?
Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$.
Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(...
- 5,529
5
votes
0
answers
471
views
A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?
In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
- 597
5
votes
0
answers
149
views
Iterative method for solving certain systems of linear equations
I've noticed that a method for calculating the stationary distribution of a finite-state rational-transition-probabilities Markov chain introduced by Arthur Engel many years ago generalizes to give an ...
- 19.7k
5
votes
0
answers
319
views
Federer's questions on the mass and comass norms
In Federer's book "Geometric measure theory", he says in section 1.8.3 (where $\|\cdot\|$ is the comass norm):
Very little appears to be known about the structure of the convex sets $\wedge^...
- 2,247
5
votes
0
answers
181
views
Is there a list of all real unital subalgebras of M(2,C)?
Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...
- 4,943
5
votes
0
answers
203
views
Number of elements in $\mathrm{GL}(n,p)$ with maximal order
I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
5
votes
0
answers
155
views
"sparsifying" a binary (over the field F2) matrix
Assume I have a matrix $A \in GF(2)$, i.e., $A_{i,j} \in \{0, 1\}$ and the sum is modulo 2. Is there any known algorithms/methods to sparsify (reduce the number of non-zero entries) $A$ while keeping ...
- 183
5
votes
0
answers
248
views
Computing powers of a special matrix fast
I've got an $n\times n$ matrix that is upper-triangular (all zeros below diagonal), diagonal entries are positive, and entries that are not on the diagonal are either $0$ or $1$. An example matrix ...
5
votes
0
answers
220
views
Rank of matrix over UFD polynomial ring
I have a matrix, $M$, size of approximately $20\times 25$, over a polynomial ring $\mathbb{Q}[x_1, \cdots, x_n]$ and a number $r$ such that I would like to test whether or not there exist values for ...
- 51
5
votes
0
answers
96
views
Partitioning the set of Pauli words into abelian pieces
Let $\sigma_x,\sigma_y,\sigma_z$ be the Pauli matrices. A Pauli word of length $n$ is defined as the tensor product $\otimes_{i=1}^n\sigma_i$ of operators $\sigma_1,\dots,\sigma_n\in\{\mathbf 1,\...
- 4,535