All Questions
6,026 questions
1
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139
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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 ...
- 111
0
votes
0
answers
129
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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^...
3
votes
0
answers
109
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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
1
vote
0
answers
39
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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 ...
- 400
0
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0
answers
32
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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
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
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(...
- 1,091
7
votes
2
answers
201
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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 ...
- 431
2
votes
1
answer
237
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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/...
- 21
9
votes
1
answer
563
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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
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 ...
- 6,351
0
votes
0
answers
77
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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
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0
answers
110
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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 \\...
- 11
0
votes
0
answers
109
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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
3
votes
0
answers
138
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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
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 ...
- 41.8k
3
votes
0
answers
212
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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:
...
- 779
2
votes
0
answers
160
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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(\...
- 593
3
votes
1
answer
193
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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 ...
- 779
4
votes
1
answer
314
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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
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 ...
- 49
6
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0
answers
130
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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 ...
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
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 \...
- 31
14
votes
2
answers
851
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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, ...
- 143
0
votes
1
answer
131
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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 } ...
- 53
5
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0
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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
2
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0
answers
78
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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 ...
6
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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
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 $A$ ...
- 41
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 ...
- 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 ...
- 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 ...
- 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 ...
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 $n^2 \cdot d$ ...
- 1,104
4
votes
1
answer
101
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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 $S$ ...
- 175
0
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0
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96
views
Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 1
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
$$
f_{\mathcal{P}}(X) = \sum_{P \in \mathcal{P}} z^T X(PX + I)^{-1} z,...
- 400
1
vote
1
answer
87
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Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?
The spectral theorem says that for every Hermitian matrix $A \in \mathbb{C}^{n \times n}$ there is a unitary matrix $V \in U(n)$ and a diagonal matrix $D \in \mathbb{R}^{n \times n}$ such that $A = ...
- 1,295
0
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0
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72
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Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
1
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0
answers
37
views
Bounding the length of an R-module of matrices
Loosely related to this: Bounding the length in a module of evaluated skew polynomials
Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
- 223
1
vote
0
answers
50
views
Interpolation in convex hull
I'm reading a paper, Learning in High Dimension Always Amounts to Extrapolation, that provides a result I don't understand.
It provides this theorem which I do understand:
Theorem 1: (Bárány and ...
3
votes
2
answers
251
views
Minimum-norm solution of $A = X B + B^T X^T$
Let $A, B, X$ be invertible square matrices, and let $A$ additionally be symmetric. I'd like to solve the following minimization problem:
$$\text{argmin}_X |\!| X |\!|_F \ \ \ \ \text{s.t.} \ \ \ \ A =...
- 453
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 ...
- 728
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 \}...
- 820
89
votes
1
answer
21k
views
Is the largest root of a random polynomial more likely to be real than complex?
This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO.
The number of real roots of a ...
- 5,470
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}=...
- 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?
- 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 ...
- 11