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Inverse of a Cauchy-like matrix

Consider $n\times n$ symmetric Cauchy-like matrix $M$ with elements $(M_{ij})_{i,j=1}^{n}$ given by $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{...
Abhishek Halder's user avatar
5 votes
1 answer
324 views

Is operadic desuspension inverse to operadic suspension?

Given a graded vector space $V$ over a field $k$, consider it's suspension $\Sigma V$ such that $(\Sigma V)^i=V^{i-1}$. For an operad of graded vector spaces over a field $\mathcal{O}$, the operadic ...
Javi's user avatar
  • 499
5 votes
1 answer
1k views

How many distinct eigenvalues does a random graph have?

It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one. But what is known about the ...
Felix Goldberg's user avatar
5 votes
0 answers
350 views

How to calculate the volume of a parallelepiped in a normed space?

Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
erz's user avatar
  • 5,529
5 votes
1 answer
993 views

Full-rank rectangular matrices over GF(2)

Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...
Robert Israel's user avatar
5 votes
0 answers
150 views

monomer-dimer tiling of a Young diagram

As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known. Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
T. Amdeberhan's user avatar
5 votes
3 answers
1k views

classifying space and cohomology of integer general linear group

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian. I have also obtained that the mod 2 cohomology is the polynomial ...
Shiquan Ren's user avatar
  • 1,990
5 votes
2 answers
542 views

Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over $\mathbb{R}$)....
Jjm's user avatar
  • 2,091
5 votes
0 answers
161 views

Where have you encountered "arrangement spaces"?

I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature). Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
M. Winter's user avatar
  • 13.6k
5 votes
2 answers
680 views

Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2).

Is there an efficient algorithm for finding the solution $x$ of $b = Ax$ that minimizes the Hamming weight of $x$, where $A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo 2")...
David's user avatar
  • 141
5 votes
2 answers
721 views

Matrices with same eigenvalues

This question is a more precise version of this question. Let's assume we have the matrix $$\left( \begin{array}{ccccc} 0 & a & 0 & 0 & 0 \\ f & 0 & b & 0 & 0 \\ 0 &...
António Borges Santos's user avatar
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} \...
T. Amdeberhan's user avatar
5 votes
1 answer
474 views

An inequality for certain positive-semidefinite matrices

Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that $$\sum_{i,j}(G^5)...
Iosif Pinelis's user avatar
5 votes
0 answers
397 views

spectrum of orthogonality graphs

The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard ...
Clive elphick's user avatar
5 votes
1 answer
325 views

The discrete Fourier transform's Gaussian-like eigenvector

I have the $N$x$N$ matrix below where $N$ is a power of 2 (usually 64 or 256) and $\omega = 2\pi/N$. What is its largest eigenvalue? $\begin{bmatrix} 2 & 1 & 0 & 0 & \cdots & 0 &...
bobuhito's user avatar
  • 1,547
5 votes
1 answer
833 views

A statement on complex polynomials

I have a feeling the following is true. Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...
Malkoun's user avatar
  • 5,215
5 votes
0 answers
234 views

A combinatorial problem

What is the largest $m\times m$ $0/1$ matrix of real rank $n$ with every square submatrix sized at least ${n^{\gamma}}\times{n^{\gamma}}$ distinct for some fixed $\gamma>0$? Upper Bounds: Number ...
user avatar
5 votes
1 answer
398 views

Asymptotically nilpotent matrices

A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$. Assume that $\mathcal{A}, \mathcal{B}$ be maximal (under ...
solver6's user avatar
  • 291
5 votes
0 answers
586 views

A minimal eigenvalue inequality

Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
Hans's user avatar
  • 2,239
5 votes
2 answers
3k views

Complex Eigenvalues of Directed Graphs

I have been computing eigenvalues of adjacency matrices for several directed (not necessarily strongly connected) graphs and one remarkable property seemed to hold (each graph that I have examined ...
042's user avatar
  • 83
5 votes
2 answers
2k views

Decomposing a matrix into a product of sparse matrices

How to study the decomposition of a square matrix into a product of sparse matrices? There are no restrictions on the number of matrices in the product, but the fewer the better.
unknown's user avatar
  • 451
5 votes
1 answer
199 views

Find the inverse of a more general matrix that is similar to the Hilbert matrix

In the last MO question , the following matrix is given: $$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$ and its inverse has been discussed. Now the problem is further extended to a more general ...
Dings's user avatar
  • 153
5 votes
2 answers
203 views

dyadically recursive matrices: Part I

Introduce the $2^{n-1}\times 2^{n-1}$ matrix $B_n$ recursively as follows: $B_1(b_1)=\begin{pmatrix} b_1\end{pmatrix}$ and $$B_n(b_1,\dots,b_n)=\begin{pmatrix} B_{n-1}(b_1,\dots,b_{n-1})& b_nJ_{n-...
T. Amdeberhan's user avatar
5 votes
1 answer
214 views

When does isometric projection respect multiplication?

Every $A \in \text{GL}_n(\mathbb{R})$ has a unique orthogonal polar factor $O_A=A(\sqrt{A^TA})^{-1}$, ( $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$see Polar decomposition). ...
Asaf Shachar's user avatar
  • 6,741
5 votes
0 answers
235 views

Riemann theta function inequality for a class of large random matrices

The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
Dierk Bormann's user avatar
5 votes
2 answers
503 views

Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$

Look at the expression $$ f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1. $$ The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
Kurisuto Asutora's user avatar
5 votes
3 answers
830 views

Integral of the entrywise square of the exponential of a matrix

Note: I posted my question on math.stackexchange but got no answer. That is why I am asking it here. Let $A$ be a $n\times n$ square matrix such that the real part of all eigenvalues are negative. ...
N. Gast's user avatar
  • 562
5 votes
1 answer
265 views

Is every matrix involution over a UFD diagonalisable?

Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$). Is every involution in $\mathrm{GL}_n(A)$ diagonalisable? This is of ...
Jérémy Blanc's user avatar
5 votes
2 answers
707 views

Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers

Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...
Gro-Tsen's user avatar
  • 32.5k
5 votes
1 answer
194 views

Subalgebras of singular matrices

Is it true that any subalgebra of singular matrices have a common null-vector? In other words, is it true that, for any subalgebra $\cal S$ of the algebra of linear operators in a finite-dimensional ...
Anton Klyachko's user avatar
5 votes
3 answers
1k views

Constant rank theorem for Banach spaces

Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
Benjamin's user avatar
  • 2,099
5 votes
2 answers
550 views

Spectral radius of a rank-1 perturbation

Suppose that $\bf A$ is an $n \times n$ matrix, and $\bf u$ and $\bf v$ are vectors. The matrix determinant lemma lets us easily compute the determinant of ${\bf A} + {\bf u} {\bf v}^\top$, while the ...
user avatar
5 votes
3 answers
3k views

Spectral properties of the LDL^T matrix factorization

Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How ...
Victor Liu's user avatar
4 votes
0 answers
1k views

Number of arrangements that contain at least 1 path from top to bottom of 2D matrix

I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white. With that information, I can easily calculate the total number of black element arrangements that exist ...
Cardstdani's user avatar
4 votes
4 answers
2k views

I want a smooth orthogonalization process

The following question is related to research I am doing on reinforcement learning on manifolds. I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
Jabby's user avatar
  • 155
4 votes
4 answers
3k views

The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \...
David's user avatar
  • 41
4 votes
1 answer
410 views

power log distance between matrices

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...
Igor Rivin's user avatar
  • 96.4k
4 votes
1 answer
202 views

Subalgebras of singular matrices (less naive version)

Is it true that, for any subalgebra $\cal S$ of the algebra of linear operators in a finite-dimensional vector space $V$ over a field, $$ \bigcap_{A\in\cal S}\ker A=\{0\}\hbox{ and } \bigcup_{A\in\cal ...
Anton Klyachko's user avatar
4 votes
0 answers
238 views

Conjectural values of some determinants involving Legendre symbols (I)

$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants $$\det\left[\Legendre{i+j}p\right]_{...
Zhi-Wei Sun's user avatar
  • 15.6k
4 votes
0 answers
284 views

Institutional approach to linear algebra

In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following Definition. An institution ...
Alec Rhea's user avatar
  • 10.1k
4 votes
0 answers
282 views

Classification of special symmetric Frobenius algebras over real vector spaces

Is there a general classification of special symmetric Frobenius algebras over real vector spaces? I know that $n\times n$ matrix algebras, the quaternions, the complex numbers, the trivial algebra, ...
Andi Bauer's user avatar
  • 3,001
4 votes
1 answer
352 views

Regarding a Paper by Paul.A Clement on Tridiagonal Matrices

In Paul.A Clement's (1959) paper: A Class of Triple-Diagonal Matrices for Test Purposes SIAM Review, Vol. 1, No. 1 (Jan., 1959), pp. 50-52 He makes the claim ...
FlamingWilderbeest's user avatar
4 votes
0 answers
124 views

Distributive lattices with periodic Coxeter matrix

Let $L$ be a finite distributive lattice and $U$ its incidence matrix with entries $u_{i,j}=1$ iff $i \leq j$ and $u_{i,j}=0$ else. Then $U^{-1}$ is the Moebius matrix of $L$ and $C_L:=- U^{-1} U^{T}$ ...
Mare's user avatar
  • 26.5k
4 votes
0 answers
2k views

eigenvalues of a symmetric tridiagonal matrix with zero diagonals

I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements): $$ \left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a &...
Xiao Junhui's user avatar
4 votes
1 answer
535 views

A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type. There is an equivalence of Gerstenhaber ...
Daniel Pomerleano's user avatar
4 votes
1 answer
313 views

A determinant of perfect square polynomials

Usually, I like working with determinants related to the Vandermonde matrix, i.e. $$\det(x_j^{i-1})=\prod_{i<j}(x_j-x_i).$$ However, I run into some unusual matrix and its determinant. Define the $(...
T. Amdeberhan's user avatar
4 votes
2 answers
473 views

Completing partial matrix to a positive definite matrix

Is the following problem known? Suppose one is given some of the entries of an $n \times n$ matrix $A$ over $\mathbb{R}$, so that the given entries are symmetric. Can one assign values to the ...
Lior Gishboliner's user avatar
4 votes
1 answer
1k views

How to solve a matrix equation with both inverses and a hadamard product?

I have a matrix equation of the form: $$ A^{-1} = B + A \circ C $$ where $\circ$ denotes the Hadamard product (i.e., $(A\circ C)_{ij} = A_{ij}B_{ij}$). How can I determine if a solution for $A$ ...
Mike Izbicki's user avatar
4 votes
1 answer
720 views

Singular value decomposition of truncated discrete Fourier transform matrix

Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that \begin{align} F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N. \end{align} What we can say about the singular value ...
Math_Y's user avatar
  • 287
4 votes
0 answers
211 views

Conjecture on convergence of iterated near-matrix square root

Here is a simple problem that has stumped me for some time; sharing with the community, as I suspect it has been solved somewhere, or is immediately implied by the correct theorem. Let $\textbf{diag}: ...
Keith Rush's user avatar

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