Skip to main content

All Questions

Filter by
Sorted by
Tagged with
7 votes
2 answers
201 views

When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?

Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space. Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...
Rahul Sarkar's user avatar
2 votes
1 answer
237 views

Geometric interpretation of trace of a linear operator

This question is really an addendum to Geometric interpretation of trace There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
Bijou Smith's user avatar
9 votes
1 answer
563 views

Peter–Weyl decomposition of a group representation rather than group algebra

Consider a finite or compact group $G$. The Peter–Weyl decomposition is usually formulated for the group algebra $\mathbb{C}[G]\simeq\bigoplus_i \operatorname{End}(V_i)$, where $V_i$ are the spaces of ...
Conifold's user avatar
  • 1,731
7 votes
1 answer
271 views

Existence of a linear map resulting in the determinant being an elementary symmetric polynomial

Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
Nathaniel Johnston's user avatar
0 votes
0 answers
77 views

Eigenvalues of N×N correlation matrices as N tends to infinity

I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero. ...
Zywoo_biu's user avatar
1 vote
0 answers
110 views

Multiplication of a matrix by sub-matrices

I have a $J \times J$ matrix $C$ that is upper triangular. Also, $C'C$ is positive definite. I also have a matrix $A$ formed by submatrices of size $J \times K$ as follows $$ A = \begin{bmatrix} A_1 \\...
LauraH's user avatar
  • 11
0 votes
0 answers
109 views

Linear independence in $\mathbb{Z}_q^n$

Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...
aleph's user avatar
  • 503
4 votes
1 answer
686 views

Who and when proved Artin's Theorem on alternative rings?

I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings). Question. When has Artin proved this theorem and where was it published for the first ...
Taras Banakh's user avatar
  • 41.8k
2 votes
0 answers
160 views

An "almost" true inequality for Hermitian matrices

Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
WunderNatur's user avatar
4 votes
1 answer
314 views

Some but not all eigenvectors mutually orthogonal

Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
Michael Hardy's user avatar
1 vote
1 answer
114 views

Sum of squares of $k\times k$ cofactors is $1$ for an orthonormal matrix [closed]

Let $n,k\in \mathbb N$ with $k\leq n$. Let $A$ be an $n\times n$ real orthonormal matrix. Fix any $k$ rows of $A$ and from there consider every possible $k\times k$ cofactors and there will be exactly ...
mathew's user avatar
  • 49
6 votes
0 answers
130 views

Bent vectors and $\pm 1$ eigenvectors with respect to non-Sylvester Hadamard matrices

A Hadamard matrix is an $n\times n$-matrix $H$ where each entry in $H$ is $\pm 1$ and where $H/\sqrt{n}$ is orthogonal. It is well-known that if $H$ is an $n\times n$-Hadamard matrix, then $n<3$ or ...
Joseph Van Name's user avatar
7 votes
1 answer
390 views

Questions on symmetric Hadamard matrices

Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal. If $A$ is a symmetric matrix, then $A = A^T$ and if $...
user369335's user avatar
1 vote
0 answers
40 views

From a constraint satisfaction problem (CSP) to a sudoku grid [closed]

one of the existing methods of solvin a sudoku grid is via constraints satisfaction (CSP), but can we do the inverse ie convert a CSP problem into a sudoku grid and then solve it ?
youssef Lmourabite's user avatar
2 votes
1 answer
215 views

Forming real positive semidefinite matrices from complex matrices

I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices. Let $Q \in \...
Mthpd's user avatar
  • 31
14 votes
2 answers
851 views

Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
Mapy Duq's user avatar
  • 143
0 votes
1 answer
131 views

Function of eigenvalues of Laplacian matrix

Let $G$ be a simple $n$-vertex graph and let $\mu_n\geq\mu_{n-1}\geq\dots\geq\mu_1$ be the eigenvalues of its Laplacian matrix, how can I find a function $$f(\mu_1,\mu_2,\dots\mu_n) \text{ such that } ...
David's user avatar
  • 53
5 votes
0 answers
583 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 ...
Shahab's user avatar
  • 429
2 votes
0 answers
78 views

Partitions of bent vectors

Let $H=\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}.$ Let $A^{\otimes N}$ denote the tensor product of the matrix $A$ with itself taken $N$ times. We say that a vector $v$ of ...
Joseph Van Name's user avatar
6 votes
0 answers
111 views

Factorization to sparse matrices

$\newcommand{\lrank}{\operatorname{lrank}}$ $\newcommand{\rank}{\operatorname{rank}}$ Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it. Now, given ...
Daniel Weber's user avatar
  • 3,319
4 votes
0 answers
87 views

Non-zero element in diagonal of cofactor matrix of symmetric 0-1 matrix with non-zero determinant, zero diagonal and odd number of rows

Is the following statement correct? Let $A$ be a symmetric 0-1 matrix with non-zero determinant, all diagonal elements equal to 0 and an odd number of rows. The diagonal of the cofactor matrix of $A$ ...
user528497's user avatar
3 votes
1 answer
286 views

Unitary transformations of Vandermonde matrices forms a smooth manifold?

The space of all Vandermonde matrices $V$ with $r$ variables and degree $n$ (as below) forms an embedded submanifold of $\mathbb{R}^{(n+1) \times r}$ when $x_{i} \in \mathbb{R}$. It is naturally a ...
patchouli's user avatar
  • 275
3 votes
1 answer
87 views

Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real

Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing ...
Ben Deitmar's user avatar
  • 1,295
2 votes
0 answers
67 views

Characteristics of conjugate gradients' iterations for a matrix with clustered spectrum

I am interested in solving \begin{equation} Ax = b \end{equation} for a large sparse linear symmetric positive definite matrix $A$ by Conjugate Gradients method. (These systems usually come as ...
Korf's user avatar
  • 121
0 votes
0 answers
19 views

Equality Issue in Deriving Covariance Update for Kalman Filter

I am currently working on deriving the Kalman Gain from the covariance of the updated state and have encountered an equality issue that I am unable to resolve. Below are the derivation steps and the ...
user528329's user avatar
9 votes
2 answers
245 views

Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$

Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...
Greg Zitelli's user avatar
  • 1,104
4 votes
1 answer
101 views

Extension of scalars for bounded chain complexes of $kG$-modules

I'm wondering if a generalization regarding a statement from Curtis-Reiner holds. The original statement is as follows: (30.33) Theorem: Let $R$ and $S$ be complete discrete valuation rings, with $S$ ...
Sam K's user avatar
  • 175
2 votes
1 answer
187 views

Maximum of $\sum_{n=1}^N z^T X(P_n X + I)^{-1}z$ over unit trace, positive semidefinite matrices?

Let $z$ denote a unit vector. Fix a finite collection of positive semidefinite matrices $\mathcal{P}$. Define the function and set $$ f_{\mathcal{P}}(X) = \sum_{P \in \mathcal{P}} z^T X(PX + I)^{-1} z,...
Drew Brady's user avatar
1 vote
1 answer
87 views

Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?

The spectral theorem says that for every Hermitian matrix $A \in \mathbb{C}^{n \times n}$ there is a unitary matrix $V \in U(n)$ and a diagonal matrix $D \in \mathbb{R}^{n \times n}$ such that $A = ...
Ben Deitmar's user avatar
  • 1,295
0 votes
0 answers
72 views

Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX

I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
usergh's user avatar
  • 43
1 vote
0 answers
37 views

Bounding the length of an R-module of matrices

Loosely related to this: Bounding the length in a module of evaluated skew polynomials Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
JBuck's user avatar
  • 223
1 vote
0 answers
50 views

Interpolation in convex hull

I'm reading a paper, Learning in High Dimension Always Amounts to Extrapolation, that provides a result I don't understand. It provides this theorem which I do understand: Theorem 1: (Bárány and ...
Christopher D'Arcy's user avatar
3 votes
2 answers
251 views

Minimum-norm solution of $A = X B + B^T X^T$

Let $A, B, X$ be invertible square matrices, and let $A$ additionally be symmetric. I'd like to solve the following minimization problem: $$\text{argmin}_X |\!| X |\!|_F \ \ \ \ \text{s.t.} \ \ \ \ A =...
dotdashdashdash's user avatar
4 votes
1 answer
342 views

rank of an integer valued matrix

I make some numerical experiments, involving rank of integer valued matrices of the size about $14\times 24$. As the matrix is integer valued, theoretically there should be no room for errors. However ...
Dmitri Scheglov's user avatar
0 votes
0 answers
38 views

Sequence of projections that alters a $2^d$ tuple of points to a hyperparallelepiped

Suppose we have a $2^d$ tuple $\{ x_i \}_{i=0}^{2^d-1}$ of points in some $\mathbb{R}^n$. I would like to shift the points of this tuple in some controlled way, so that the final $2^d$ tuple $\{ y_i \}...
Kacper Kurowski's user avatar
1 vote
0 answers
64 views

Reference request for non-banded Toeplitz matrix

I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix. I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
hos's user avatar
  • 11
3 votes
0 answers
153 views

Categorification of a vector space such that a functor between these is a linear map?

A functor between two monoids seen as 1 object categories is essentially a monoid-homomorphism. What is the equivalent construction for vector spaces and linear maps?
rick's user avatar
  • 199
1 vote
0 answers
66 views

Characterising lattices $\Lambda\subseteq\mathbb{Z}^n$ whose union of translations by $b\in\{0,1\}^n$ recovers $\mathbb{Z}^n$

Given a lattice $\Lambda\subseteq\mathbb{Z}^n$ defined by $\Lambda = \{ Mx : x\in\mathbb{Z}^n \}$, let $\Lambda_b$ for $b\in \{0,1\}^n = B$ be the translation of $\Lambda$ by $b$. Call $M$ special ...
mr bean's user avatar
  • 11
5 votes
1 answer
196 views

What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?

Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
AlpinistKitten's user avatar
1 vote
0 answers
60 views

Bounding the length in a module of evaluated skew polynomials

Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
JBuck's user avatar
  • 223
6 votes
1 answer
244 views

Linear independence over field of rational functions

To prove that functions $f_1(x), \dots, f_n(x)$ with $x \in \mathbb R$ are linearly independent, we only need to show that the Wronskian of these functions is non-zero at a certain value of $x$. Now ...
Pluviophile's user avatar
  • 1,608
1 vote
0 answers
45 views

Rank of Hadamard product of column-wise polynomial evaluations and row-wise exponential evaluations

Consider the Hadamard product $A \odot B$ between two special matrices $A,B \in \mathbb{R}^{n \times m}$. The columns of $A$ are evaluations of polynomials, while the rows of $B$ are evaluations of ...
user31127's user avatar
4 votes
0 answers
108 views

Larger possible chain of closed subspaces in the dual of a Banach space

In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces. My question is the following. If $X$ is an ...
Emerick's user avatar
  • 153
1 vote
0 answers
195 views

Conjectural values of some determinants involving Legendre symbols (II)

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by the evaluation of the determinants $$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{...
Zhi-Wei Sun's user avatar
  • 15.6k
0 votes
0 answers
123 views

Realizable singular value spectra of normalized finite frames

$\DeclareMathOperator\tr{tr}$Let $m, n \in \mathbb{N}$, $m \geq n$, and let $\{f_i\}$, $1 \leq i \leq m$, be $m$ unit vectors (wrt. 2-norm) in $\mathbb{R}^n$. Let $A = [f_1 \, \, \, f_2 \, \, \, \...
J. Zimmermann's user avatar
0 votes
0 answers
36 views

Conjugate gradient-like algorithm with multiple search directions

I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm. I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
SRB121's user avatar
  • 71
0 votes
1 answer
60 views

Symmetric positive definite matrix - submatrices

Let $\lambda$ and $\Lambda$ be two fixed positive numbers and let $A$ be a symmetric real matrix with $\lambda |x|^2 \leq (Ax,x) \leq \Lambda |x|^2$. Let $B$ be a matrix derived from $A$ as follows: $...
Adi's user avatar
  • 455
0 votes
1 answer
317 views

A variation of the Riesz Lemma

Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
Emerick's user avatar
  • 153
2 votes
1 answer
345 views

What's the explicit value of this determinant

Let $n\ge2$ be a positive integer, and let $b_1,\cdots,b_n, c_1,\cdots, c_n$ be variables. Recently, I met the following determinant: $$\det A=\left|\begin{array}{cccc} 1 & b_1+c_1 & b_1^2+c_1^...
Beginner's user avatar
1 vote
0 answers
189 views

The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$

There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
lunch zheng's user avatar