All Questions
6,178 questions
1
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175
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Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
3
votes
1
answer
99
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Eigenvectors of $P^\top P$ for 0/1 matrices $P$
Let $P$ be an $m \times N$ matrix of zeros and ones (think $N \gg m$), and let $\mathbf{u} \in \mathbb{R}^N$ be a unit vector satisfying $P^\top P\mathbf{u} = \lambda^2 \mathbf{u}$ for some $\lambda &...
- 161
3
votes
1
answer
252
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Two isotropic subspaces in a symplectic vector space
Let $k$ be a field of characteristic $0$, let $V$ be a finite-dimensional vector space over $V$, and let $\omega(-,-)$ be a symplectic bilinear form on $V$. In other words, $\omega(-,-)$ is an ...
- 33
3
votes
1
answer
189
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Rank properties of matrix valued in linear forms
Let $R(X,Y) \in \text{Mat}_{d,d+1}(\mathbb{C}[X,Y]_{(1)})$ be a $d \times (d+1)$-matrix valued in linear forms $\mathbb{C}[X,Y]_{(1)}:= \{aX+bY \ \vert \ a,b \in \Bbb C \}$.
Let denote $v_j(X,Y)$ its $...
- 6,018
8
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4
answers
379
views
Traceless Hermitian matrices with simultaneously vanishing Rayleigh quotients
Let $D$ be an integer greater than 1. What is the largest number $N$, such that for all sets of $N$ Hermitian $D\times D$ traceless matrices $M_i$, $i=1,\dots,N$, there exists a non-zero complex ...
- 628
0
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0
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43
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Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_\text{min}$, of $A$?
What I want is something like: $\sigma_\...
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
1
vote
1
answer
391
views
How one can show that this matrix is full rank?
Fix $d\in\mathbb{N}$ and consider $e_{i,j}\in\mathbb{C}$ for $i=1,\dots,d+3$ and $j=1,\dots,d-1$. Suppose to have the following matrices
$$N_{i,1}=\begin{pmatrix}
1 & 0 \\
e_{i,1} & 1
\end{...
- 11
0
votes
0
answers
40
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Eigendecomposition of Toeplitz matrix
I am working with Toeplitz matrix, I know that a Toeplitz Matrix $T$ can be decompose as a sum of a circulant and skew circulant matrix which can be diagonalized using the DFT matrix
$T=C+S = F\...
- 1
5
votes
1
answer
279
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 541
1
vote
1
answer
145
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Prove or disprove that the matrix equation of the form $AX+XA^{-T}=0$ has a nonsingular anti-symmetric solution $X$
I’m trying to prove that for $A=J_n(i)$, that is, the Jordan block matrix corresponding to the eigenvalue $i$ of size $n$, where $n$ is even, the matrix equation $AX+XA^{-T}=0$ has a nonsingular anti-...
- 11
3
votes
1
answer
196
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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 ...
- 183
1
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0
answers
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
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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
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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 ...
- 380
0
votes
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
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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
9
votes
1
answer
553
views
Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here.
Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
- 541
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 ...
- 431
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/...
- 21
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
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1
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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
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
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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 \\...
- 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
4
votes
1
answer
685
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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
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(\...
- 593
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
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
votes
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
388
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
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0
answers
40
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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
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1
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214
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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
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, ...
- 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
votes
0
answers
582
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
votes
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
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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
9
votes
1
answer
845
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Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
- 541
4
votes
0
answers
87
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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 $n^2 \cdot d$ ...
- 1,084
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 $S$ ...
- 175
11
votes
2
answers
587
views
Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
- 271
6
votes
2
answers
380
views
Proving convergence of solution of a fixed point equation
I encountered a nasty sequence $(x_n)_{n=1}^\infty $ defined as the smallest positive fixed point of the fixed point equation $ x_n = f_n(x_n) $, where $f_n$ is given by
$$ f_n(x) = \sum_{k=0}^{\...
- 61