All Questions
5,883 questions
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Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
CommunityBot
- 1
1
vote
0
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64
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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
4
votes
1
answer
342
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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 ...
- 3,319
0
votes
0
answers
38
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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 \}...
- 2,858
3
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0
answers
153
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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
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0
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66
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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
1
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0
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195
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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
$$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{...
CommunityBot
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1
vote
1
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297
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Nearest Kronecker product to sum of Kronecker products
I am interested in efficiently finding the closest Kronecker decomposition to the sum of $k$ Kronecker products:
$$\min_{A,B} || A \otimes B - \sum_{i=1}^k A_i \otimes B_i ||_F$$
where $A,A_i$ are $p \...
- 111
4
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0
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108
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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 ...
- 153
6
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1
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244
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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 ...
- 91.8k
1
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0
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60
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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 \...
- 223
1
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0
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45
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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 ...
- 21
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0
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123
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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 \, \, \, \...
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1
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125
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Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
- 108k
1
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1
answer
99
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Maximum column norm of random $A^{-1}B$
Suppose that $A$ is an $n$ by $n$ Gaussian matrix (each component i.i.d. normal distributed with mean 0 and variance 1). Let $b$ be a $n$-Gaussian vector. Then it could be easily proven that the ...
- 3,741
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0
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36
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Conjugate gradient-like algorithm with multiple search directions
I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm.
I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
- 71
9
votes
3
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861
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A curious equation on determinant----linear algebra or algebraic geometry?
I recently find a curious and unexplainable(as seems to me) equation on determinant as follows.
$$3\begin{vmatrix}
a_1 & b_1 & c_1 & d_1 \\
a_2 & b_2 & c_2 & d_2 \\
...
- 281
2
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1
answer
345
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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:
$$\det A=\left|\begin{array}{cccc}
1 & b_1+c_1 & b_1^2+c_1^...
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1
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317
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A variation of the Riesz Lemma
Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
- 60.5k
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1
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60
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Symmetric positive definite matrix - submatrices
Let $\lambda$ and $\Lambda$ be two fixed positive numbers and let $A$ be a symmetric real matrix with $\lambda |x|^2 \leq (Ax,x) \leq \Lambda |x|^2$.
Let $B$ be a matrix derived from $A$ as follows: $...
- 128k
1
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0
answers
189
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The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 30.1k
4
votes
0
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238
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Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
0
votes
0
answers
47
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Counting zero-sum subsets of a finite field with a particular form
Let $\mathbb{F}$ be a finite prime field of characteristic different than $2$ and $\beta \in \mathbb{F}$ a generator of the $2$-power order multiplicative subgroup of order $2^k$, so $\beta^{2^{k-1}} =...
- 3,058
1
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0
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255
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Interpreting positive semidefinite matrix as a graph
Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
CommunityBot
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1
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184
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Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?
Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
- 24.9k
10
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2
answers
4k
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The log transform turns scalar multiplication into addition. Is there an analogous transformation for matrix-vector multiplication?
Napier's method of logarithms and corresponding tables of logarithms provided a important tool to simplify hand computation by converting multiplication and division to equivalent problems of addition ...
CommunityBot
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3
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2
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215
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Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$
Is there a closed-form solution for
$$\max_D \operatorname{Tr}(ADBD)$$
where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...
3
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2
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453
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Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 2,178
0
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1
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100
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Projection on a countable union of linear subspace
For any natural number $n$, $V_n$ denotes a closed linear subspace of a $L_2(m)$ space, which is an Hilbert Space, where $m$ denotes a finite measure. Moreover $(V_n)$ is increasing, that is $V_n$ is ...
- 128k
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0
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66
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Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
0
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1
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127
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update rule for the inverse after a rank-1 update plus scaled identity
Is there an update rule for $$\left(\tilde{X}^T\tilde{X}+\alpha\cdot I\right)^{-1}$$ with $\tilde{X}=[X\;\; a]$ as a function of $A\triangleq (X^TX)^{-1}$, $X$ and $a$?
I know that when $\alpha=0$ we ...
- 20.2k
3
votes
0
answers
105
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Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
- 63.9k
2
votes
2
answers
1k
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Rank of a linear combination of quadratic forms
Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
- 50.6k
0
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0
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28
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Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
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80
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Inequality involving minors of an orthogonal matrix
Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
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2
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1
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170
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Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression
Consider the multivariate regression model
$$Y = XB + E$$
where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...
5
votes
1
answer
728
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Linear independence of exponentials
Let $X$ be the set of functions $e^{p(x)}$ of the real vector $x$, where $p$ is a multivariate polynomial with $p(0)=0$.
Is any finite subset of $X$ linearly independent? If yes, why? If no, is the ...
- 63.9k
1
vote
0
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138
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Questions on integer matrix multiplication
Question 1:
Given two integer matrices $A$ and $B$, and let $C$ be $AB$.
$C$ can be very big in pratice, so what is the fastest way to compute the statistical data of $C$?
For example,
$$A=\begin{...
- 696
5
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3
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560
views
An inequality in an Euclidean space
For $n\geq 1$, endow $\mathbb{R}^n$ with the usual scalar product. Let $u=(1,1,\dots,1)\in\mathbb{R}^n$, $v\in {]0,+\infty[^n}$ and denote by $p_{u^\perp}$ and $p_{v^\perp}$ the orthographic ...
- 3,599
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2
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397
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How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 24.2k
4
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0
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99
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If matrices $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup \mathbb{T}$, does $\text{rank}([A,B])=\text{rank}([A^*,B])$?
Here $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$ denotes the unit circle in the complex numbers.
This holds, if we have $\sigma(A)\subseteq \mathbb{R}$ or $\sigma(A)\subseteq \mathbb{T}$ (independent of $B$...
- 151
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0
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81
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Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?
I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
0
votes
1
answer
103
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Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
CommunityBot
- 1
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0
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32
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Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
1
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0
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29
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Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?
Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
- 11
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1
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512
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Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
CommunityBot
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7
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0
answers
225
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Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 11.1k
5
votes
1
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349
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Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity
Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $\mathrm{GL}_n(\mathbb{Z})$, $\forall n\geq 0$. I have ...
- 7,893
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0
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36
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A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial
We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle.
Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
- 4,058
3
votes
1
answer
240
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Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$
Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. We find an upper triangular basis matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ of $\Lambda$. Is $B$ unique up to the right action of $\...
- 20.2k