All Questions
6,026 questions
2
votes
0
answers
75
views
Smallest dimension, on which a set of matrices acts non-trivially
Let $A_i$, $i=1,\dots,N$, be a finite set of $D<\infty$ dimensional Hermitian matrices. Let $d$ be the smallest number for which there exists a unitary $D$-dimensional matrix $U$, and Hermitian $d$-...
- 628
0
votes
0
answers
40
views
Eigendecomposition of Toeplitz matrix
I am working with Toeplitz matrix, I know that a Toeplitz Matrix $T$ can be decompose as a sum of a circulant and skew circulant matrix which can be diagonalized using the DFT matrix
$T=C+S = F\...
- 1
3
votes
1
answer
196
views
Deriving the "Explicit" formula for inverse of Hilbert/Cauchy matrices
My exact question is, how to derive the formula for $H^{-1}$, in which $H_{ij}=\frac{1}{i+j-1}$.
I am currently working my way through Hoffman&Kunze Linear Algebra. I noticed that a question on ...
- 12.9k
2
votes
2
answers
381
views
(Non-topological) interior of a convex set
Given a convex subset $X$ of a real vector space $V$, I'm interested in the set $$Y:=\{x\in X:\ \forall v\in V, \ \exists\epsilon>0 \text{ s.t. } x+\epsilon v \in Y \}.$$
My question is boring: ...
CommunityBot
- 1
1
vote
0
answers
139
views
Integral convex polytopes formed from the weight diagrams of representations of $\mathfrak {sl}_4$($\mathbb{C}$)
I'm a student studying undergraduate abstract algebra and doing a summer research project in the mathematics department at my school. I'm barely familiar with the rudiments of representation theory; I ...
- 63.9k
20
votes
6
answers
42k
views
Eigenvalues of symmetric tridiagonal matrices
Suppose I have the symmetric tridiagonal matrix:
$$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & \ddots & \vdots \\\
0 & b_{2} & a & \...
- 21
0
votes
0
answers
129
views
Linearly independent Kronecker product construction
I have a question regarding a constructive argument about Kronecker products which came up while trying to solve a more general problem.
Let $n\in \mathbb{N}$ and $E \subseteq [n] \times [n]$ with $d^...
1
vote
0
answers
39
views
Constructing a centered distribution absolutely continuous with respect to uniform measure on the sphere with a pre-specified covariance?
Let $\mathcal{K}_n$ denote the space of $n \times n$, real symmetric positive semidefinite matrices $K$ having unit trace.
It is easy to verify that for each $K \in \mathcal{K}_n$, there exists a ...
- 410
3
votes
0
answers
109
views
How much a general a theory of matrices equivalence under group actions we have?
Let $F$ be a field and let $M_{m,n}\,(F)$ be the $F$-linear space of $m \times n$ matrices over $F$. Let $G$ be a group acting on $M_{m,n}\,(F)$.
My question is: Do we have some theory about the ...
- 157
0
votes
0
answers
101
views
Eigenvectors of tridiagonal hermitian matrix
In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as:
\begin{align*}
Q_n =
\begin{pmatrix}
0 & q_{1,2} & 0 & 0 & ...
- 1
0
votes
0
answers
32
views
Finding measure representation for rank 2 moment matrices
Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it.
$$
\begin{bmatrix}
1 & 1 \\
...
- 275
3
votes
1
answer
102
views
Literature containing basic knowledge of homogeneous functions
Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
- 63.9k
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 $...
- 696
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/...
- 118k
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 ...
- 6,351
368
votes
31
answers
80k
views
Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...
- 869
3
votes
0
answers
212
views
Differentiability along hyperplanes for rational functions
This is a follow up to my previous question.
Let $f\colon \mathbb R^3\to \mathbb R$ be a continuous function that is rational and differentiable along all planes through $0$, that is, we assume:
...
CommunityBot
- 1
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 ...
- 33.8k
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 ...
- 1,731
7
votes
3
answers
5k
views
Proof for a rank-one decomposition theorem of positive (semi) definite matrices
$\DeclareMathOperator\rank{rank}\DeclareMathOperator\trace{trace}$Consider the following result which I recently came across in a research paper in my area (signal processing)
Let $X$ be a $N\times N$...
- 63.9k
7
votes
2
answers
1k
views
Why is the spectrum of Erdős–Renyi random graph approximately symmetric?
I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős–Renyi random graph $G(n,p)$ when $np\to\infty$,
and $np\to c=2,3$.
The plots above are already ...
- 5,405
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 $n^2 \cdot d$ ...
- 21.5k
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 \...
- 63.9k
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 ...
- 785
0
votes
1
answer
269
views
Product of subspace and its inverse
$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
CommunityBot
- 1
7
votes
2
answers
604
views
Minimizing the largest eigenvalue of random matrices
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix with entries $A_{ij} \sim \mathcal{N} (0,1)$, all independent except for the symmetry condition.
Consider the following minimization problem:...
CommunityBot
- 1
-1
votes
1
answer
825
views
How to calculate determinants of such types?
Consider next determinant that we want to expand around $h=1$
\begin{eqnarray}
Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
CommunityBot
- 1
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 ...
- 1,422
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.
...
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
$$
A = \begin{bmatrix} A_1 \\...
- 10.4k
0
votes
0
answers
109
views
Linear independence in $\mathbb{Z}_q^n$
Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...
- 503
4
votes
1
answer
686
views
Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 188k
2
votes
0
answers
160
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\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(\...
- 5,215
3
votes
0
answers
138
views
What is the probability that the absolute value of the root of a polynomial is greater than $x$?
Note: This question was unanswered in MSE for a month so posting it in MO.
Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we ...
- 5,470
3
votes
1
answer
193
views
Differentiability along hyperplanes
Definition. Let us say that a function $f\colon \mathbb R^d\to \mathbb R$ is differentiable along hyperplanes in the point $0\in \mathbb R^d$, if $f\circ \varphi\colon \mathbb R^{d-1}\to \mathbb R$ is ...
- 10.6k
1
vote
1
answer
452
views
About the Hadamard conjecture
On the wikipedia article about Hadamard Matrix it says that "The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is $92$"
But it also says that ...
- 696
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 ...
- 429
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 ...
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 =...
- 20.2k
3
votes
4
answers
551
views
How big a class of lines can a non-linear transformation map to itself?
Edit: In the original version of this question, I wrote "lines through the origin" instead of "lines"; as Alexandre Eremenko points out in his answer, this makes the question too ...
- 7,893
2
votes
2
answers
286
views
Inequality on numerical range of inverse of kernel matrix
Let $k(.,.)$ be a function that takes two vectors as input and outputs a scalar as follows
\begin{align}
\mathcal{k}(x,y) = \exp\left(-\frac{\|x-y\|_2^2}{2}\right)
\end{align}
where $\|x\|_2$ denotes ...
- 21
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 ...
- 108k
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 ...
- 3,319
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, ...
CommunityBot
- 1
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 ?
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 \...
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 } ...
- 34.3k
4
votes
2
answers
982
views
Simultaneous decomposition into generalized eigenvectors
This is my first question here, so please excuse me if it is too elementary.
I was wondering if the notion of a simultaneous decomposition into eigenspaces could be generalized in a special way I ...
- 1,565
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 ...
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,...
- 360