All Questions
921 questions with no upvoted or accepted answers
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
$$\mathbf{K} = \...
3
votes
0
answers
148
views
Linear combinations of special matrices
I am a hobby computer scientist and I have a problem to which I am searching an efficient algorithm.
Given an integer n, we want to combine some square input-matrices of size n in a way that is ...
- 73
3
votes
0
answers
49
views
Stability of matrix equation
Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
- 435
3
votes
0
answers
2k
views
Multiplication of two Pauli string
Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $
Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $.
Here $I,X,Y,Z$ are Pauli matrices defined explicitly as:
$$
I = \begin{...
- 131
3
votes
0
answers
111
views
Infinite ordered products (reference request)
While writing arXiv:1510.05757v2, I found myself proving some basic facts about products of Banach algebra elements over an infinite totally ordered set. (Statements and proofs are in Appendix C.) The ...
- 2,284
3
votes
0
answers
148
views
Spectrum of symmetric Toeplitz matrix
A matrix is Toeplitz if it is constant on the diagonals parallel to the main diagonal.
I am looking for references on the spectrum of finite symmetric Toeplitz matrices over finite fields.
- 221
3
votes
0
answers
56
views
Is the outer automorphism group of a finite poset finite when the Coxeter matrix has finite order?
Let $P$ be a finite connected poset.
The Cartan matrix $C_P$ of $P$ is defined as the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else for $i,j \in P$.
The Coxeter matrix of $P$ is ...
- 26.5k
3
votes
0
answers
374
views
Eigenvalues of block matrix
Given scalars $\alpha, \beta \in \mathbb{R}$, a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ and a flat matrix $B \in \mathbb{R}^{m\times n}$, where $m < n$, can I say ...
- 125
3
votes
0
answers
327
views
Homology $H_{\ast}(T, V)$
Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$; i.e.
$V:=\left\{\left(
...
- 259
3
votes
0
answers
138
views
Is there a method to solve a non-linear quadratic matrix equation?
I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$
Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
- 31
3
votes
0
answers
47
views
Factorization of a bilinear matrix-valued function
Suppose that $F(u, v) = \sum_{i}\sum_j u_i v_i C_{ij}$ is a bilinear matrix-valued function,
where $C_{ij}$ are known matrices.
Is there a relatively easy way to factorize $F$ so that
the $u$ and $v$ ...
- 31
3
votes
0
answers
108
views
Positive vector in the kernel of an skew-symmetric incidence matrix
Let $G=(V,A)$ be an oriented graph, stronlgy connected with $n\in\mathbb{N}^*$ vertices. Let $M\in\mathcal{M}_n(\mathbb{R})=(m_{i,j})$ be an skew-symmetric matrix of size $n$ and rank $r$, such that ...
- 449
3
votes
0
answers
155
views
Frobenius inner product of a zero line-sum matrix and a doubly stochastic matrix
Let $A$, $B$ be two $n\times n$ real matrices.
Let $A$ be a zero line-sum matrix where each row sum and each column sum equals zero, i.e., $$\sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{n}a_{ij}=0 $$ (it seems ...
- 71
3
votes
0
answers
160
views
Euclidean volume of symmetric matrices in operator norm
This is a nearly identical question to Euclidean volume of the unit ball of matrices under the matrix norm except in the symmetric case.
Let $\mathrm{Sym}_{n \times n}(\mathbb{R})$ be the space of ...
- 199
3
votes
0
answers
39
views
A non-singularity property for sets of real matrices
Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
- 943
3
votes
0
answers
156
views
Left and right topological K-theory of Banach algebras
Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette.
The definition is based on ...
- 356
3
votes
0
answers
1k
views
Rank of Vandermonde matrices
Consider a Vandermonde matrix
$$V = \begin{bmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\
1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\
& & \vdots \\
1 & x_n &...
- 297
3
votes
0
answers
75
views
Solutions to a special confluent Vandermonde system
Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define
$$
M^{(0)} = \begin{pmatrix}
1 &...
- 613
3
votes
0
answers
44
views
Cartan matrices of Schurian algebras
A finite dimensional quiver algebra $A$ is called Schurian in case between two points $i$ and $j$ there is at most one path. Equivalently the Cartan matrix $C_A$ of $A$ has only entries 0 or 1. Here ...
- 26.5k
3
votes
0
answers
270
views
How to compute a simultaneous block-diagonalization?
Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $a \in S$ then $a^* \in S$). The algebra generated by $S$ is a finite dimensional $*$-...
3
votes
0
answers
231
views
Singularity of symmetric block matrix with singular diagonal blocks
One can show that the following statement holds:
Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank,
$$\begin{bmatrix}A & B \\ B^T &...
- 73
3
votes
0
answers
97
views
Minimal localization need it to "diagonalize" a matrix
Let $A$ be an $n\times n$-matrix over $\mathbb Z[t^\pm]$. In general doesn't exist $P,Q\in GL(n,\mathbb Z[t^\pm])$ such that $PAQ$ is a diagonal matrix (this happens cause $\mathbb Z[t^\pm]$ is not a ...
3
votes
0
answers
255
views
Homotopicity of $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ as morphisms from $A$ to $A\otimes A$
let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.
Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\...
- 356
3
votes
0
answers
111
views
Approximate inverse of large sparse matrix
Given a large sparse matrix $M$, how to determine the existence of a good preconditioner? In other words, does there exist a sparse matrix $X$ such that $X M$ is close to the identity with respect to ...
- 31
3
votes
0
answers
165
views
A combinatorial / geometric interpretation of compositional inversion via matrix inversion
There are several ways of finding the power or Taylor series for the compositional inverse of a function $f(x)$ with $f(0)=0\;$ given its series expansion, e.g., by using the classic Lagrange ...
- 10.5k
3
votes
0
answers
89
views
The rank of a special matrix
Suppose that $P$ is a polynomial of degree $d:=\deg P$ over a field $\mathbb F$ of zero characteristic, splitting completely into pairwise distinct linear factors, and $B,C\subset\mathbb F$ are sets ...
- 23k
3
votes
0
answers
180
views
Automorphisms of infinite matrix algebra
This is a similar question to one that I posted in MSE a few days ago.
I recently came across this paper from Alahmedi, Alsulami, Jain and Zelmanov, which quoted the following result for $M_\infty(K)$...
- 195
3
votes
0
answers
359
views
Do we know what the impulse to "introduce" the Jordan canonical form was?
Mo-ers,
Do you know how it was that the study of the Jordan canonical form began?
There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
- 87
3
votes
0
answers
77
views
How can I find the integral orthogonal group of a given symmetric positive definite form?
I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix:
$$M=\begin{bmatrix}
x_1 &...
- 362
3
votes
0
answers
122
views
Algebra of block matrices with scalar diagonals
I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
- 3,041
3
votes
0
answers
65
views
How to show that a continuous family of symmetric matrices is uniformly positive?
My problem : I have a family of $4 \times 4$ symmetric matrices. More precisely consider an interger $d$, a real $\lambda> 0$ and define the family $S_{\lambda}$:
$ \{A(\lambda,x_1,x_2) ; (x_1,...
- 31
3
votes
0
answers
112
views
Similar reduced integral matrices
Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...
- 1,766
3
votes
0
answers
252
views
On the existence of fixed points of a matrix iteration
Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...
- 2,712
3
votes
0
answers
62
views
How likely is a matrix with exactly $n$ number $1$s per row to avoid a large wide empty submatrix?
Consider the finite collection $M(N,n)$ of all $N \times N$ matrices with exactly $n$ entries per row equal to $1$ and all other entries equal to zero $0$.
By an $a \times b$ submatrix of $M$ we ...
- 1,955
3
votes
0
answers
117
views
Sparsest similar matrix
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...
3
votes
0
answers
178
views
On a matrix inequality based on the Schur-Horn theorem
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.)
Let $A_s$ denote the symmetric part of $A$...
- 2,712
3
votes
0
answers
244
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
3
votes
0
answers
499
views
Eigenvectors of sum of SO(3) matrices
I asked this question before on MSE but go no answers. It seems that the problem is rather difficult so I thought of trying here. Given two matrices $A,B\in SO(n)$, each describing a rotation by ...
- 139
3
votes
0
answers
82
views
Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?
$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
- 6,741
3
votes
0
answers
125
views
An eigenvalue of certain family of matrices
Consider the matrices
$$M_n=\left[\binom{i}j+\binom{2n+1-i}{j-i}+\binom{2n+1-i}j\right]_{i,j=0}^n.$$
I am convinced and hence would like to ask:
Question: Is $0$ an eigenvalue of $M_n$?
- 43.2k
3
votes
0
answers
489
views
Generalization of Carleman coefficients to multivariable functions - Carleman tensor?
Recently I learned about a matrix called
Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.
Carleman linearization is a technique used to embed a finite
...
- 133
3
votes
0
answers
220
views
Simultaneous Congruence of Two Matrices
Could you please let me know an answer to the following question on simultaneous congruence of two matrices. This question came up while trying to handle a system of PDE.
QUESTION:
Let $A,B\in M_n(...
- 895
3
votes
0
answers
75
views
Group generated by symmetric shears
Consider the multiplicative group generated by matrices of the form
$$
\begin{bmatrix}
{1} & { 0} & { c_1} & {c_3} \\
{0} & {1} & {c_3} & {c_2} \\
{0} & {0} &...
- 1,010
3
votes
0
answers
128
views
Representation of a matrix ring
Years ago, I read a paper about how to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequences of two simpler matrices $S$ ...
- 475
3
votes
0
answers
629
views
Diagonal elements of Hermitian matrices with given eigenvalues
Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...
- 31
3
votes
0
answers
82
views
Maximum number of negative entries in a matrix with positive diagonal and given rank
Suppose $A \in \mathbb{R}^{n \times n}$ has positive entries on it main diagonal and $\mbox{rank}(A) =: d < n$. Then, what is the maximum number of many negative entries $A$ can contain?
If, in ...
- 31
3
votes
0
answers
151
views
Largest eigenvalue divided by $n$
Let $X$ be an $n\times n$ symmetric random matrix whose diagonal is fixed as $1$, and every element in the upper triangle (excluding the diagonal) is drawn from Bernoulli($p$). The elements in the ...
- 272
3
votes
0
answers
414
views
Eigenvalue distribution of a special symmetric matrix of uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
all other elements in the upper triangle are uniform random variables over $[0,1]$. all ...
- 272
3
votes
0
answers
83
views
Particular decomposition of $SU(n)$
Given $a,b \in \mathfrak{su(n)}$ which generate the full algebra, it is possible to write and $G \in SU(n)$ as:
$G = \exp(\alpha_1 a)\exp(\beta_1 b) \ldots \exp(\alpha_m a)\exp(\beta_m b)$
for some ...
- 2,099
3
votes
0
answers
104
views
Rank relation to maximum subpermanent and subdeterminant?
Given a $\pm1$ matrix $M$ of rank $r$ let the largest subdeterminant be $d$ and let the largest subpermanent be $p$.
Are there relations/bounds that connect $r$, $d$ and $p$?
Are there geometric and ...
- 13.9k