All Questions
6,176 questions
0
votes
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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
votes
1
answer
127
views
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
views
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
views
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
votes
0
answers
28
views
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
2
votes
0
answers
80
views
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 ...
- 21
2
votes
1
answer
170
views
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
views
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
122
votes
5
answers
27k
views
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 1,016
1
vote
0
answers
138
views
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,
$$A=\begin{...
- 696
5
votes
3
answers
560
views
An inequality in an Euclidean space
For $n\geq 1$, endow $\mathbb{R}^n$ with the usual scalar product. Let $u=(1,1,\dots,1)\in\mathbb{R}^n$, $v\in {]0,+\infty[^n}$ and denote by $p_{u^\perp}$ and $p_{v^\perp}$ the orthographic ...
- 3,599
0
votes
2
answers
397
views
How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
- 24.2k
4
votes
0
answers
99
views
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
2
votes
0
answers
81
views
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
views
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
0
votes
0
answers
32
views
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
vote
0
answers
29
views
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
2
votes
1
answer
286
views
Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?
Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
- 109k
2
votes
1
answer
512
views
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
- 1
7
votes
0
answers
225
views
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
answer
349
views
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
0
votes
0
answers
36
views
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
views
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
3
votes
3
answers
979
views
Determinant of Jacobian and directional derivatives
I have a function $f: \Re^2 \to \Re^2$ and would like to understand why
$$|Jf(x)|=\max_\theta|D_\theta f(x)|\cdot\min_\theta|D_\theta f(x)|$$
that is, why the determinant of the Jacobian of $f$ at $...
6
votes
1
answer
239
views
Attempts to define a matrix exponential over (as much as possible) general fields
Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...
- 20.2k
1
vote
0
answers
95
views
Vandermonde-type factorization of moment matrix?
Consider $n,d \in \mathbb{N}_{>0}$, there are many functions $y:\mathbb{N}^{n} \to \mathbb{R}$. Now for simplicity, we denote $y(\alpha)$ to be $y_{\alpha}$. Let $|\alpha| = \sum_{i=1}^{n}\alpha_{i}...
- 275
3
votes
0
answers
106
views
Bijectivity of a linear map between symmetric polynomials of even degree
Let $\mathfrak S_n$ be the symmetric group of permutations of $n$
letters and let $S = \sum_{\sigma\in\mathfrak S_n} \sigma$ be the
symmetrization operator.
Let $\Lambda_n^r$ be the vector space of ...
- 5,822
0
votes
0
answers
68
views
Meaning of $\langle M,\,M^{-1}\rangle$
For an $n\times n$ complex matrix $M$, is there a name for the expression $\langle M,\,M^{-1}\rangle$, where the inner product is the Frobenius one, $\langle A,\,B\rangle=\text{tr}(A^*B)$? Is there a ...
- 479
1
vote
2
answers
66
views
Distribution of the constraint matrix conditioned on the solution of the linear system
Suppose that A is a random matrix in $R^{n\times n}$, with each component independently and identically distributed (iid) according to $\mathcal{N}(0,1)$. Additionally, b is a random vector in $R^n$, ...
- 128k
2
votes
0
answers
60
views
Basis vectors using anti-commuting operators?
Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
- 1,305
0
votes
0
answers
55
views
Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
5
votes
2
answers
420
views
Maximum determinant of binary matrices with special properties
Let $A$ be an $n$-by-$n$ binary matrix (all elements are in $\{0,1\}$). It is known that the determinant of $A$ is bounded by $O(n^{n/2} / 2^n)$. I am looking for tighter upper bounds for matrices ...
- 3,549
0
votes
0
answers
43
views
Given two rectangular matrices and they yield the same results when they are multiplied by their own transposes. What can we say about them?
Suppose we have $MM^T = NN^T$, where $M$ and $N$ are both $n$ by $d$ matrices. Assume that $n$ is (much) larger than $d$, are there anything we could conclude about $M$ and $N$, aside from that $N$ ...
- 9
4
votes
1
answer
7k
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Matrix optimization problem
This is (probably) an easy one:
Given a positive definite matrix $M$, find the positive definite matrix $X$, which minimizes $\textrm{tr}(X M)$ subject to $\det(X) = 1$.
Looking for how to find X, ...
- 1,446
3
votes
1
answer
547
views
Why is this polynomial factorizable? [closed]
I met a curious problem on factorizing a homogenerous polynomial of degree 9.
Problem: Show that the following polynomial can be divided by $(a_1+a_2+a_3)$:
\begin{align}
&\quad\left|
\begin{array}...
- 357
1
vote
0
answers
37
views
When does an optimal input sequence for a discrete-time system exist?
Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...
1
vote
0
answers
73
views
What is the closed form of a polyhedral cone's dual cone?
A polyhedral cone can be defined as
$$
\mathcal{K} = \{x~|~Ax\preceq 0\},
$$
where $A \in \mathbb{R}^{m \times n}$, $x\in \mathbb{R}^n$ and $\preceq$ denotes component-wise less than and equal to.
The ...
- 71
0
votes
0
answers
64
views
When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
- 63.9k
0
votes
0
answers
28
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Find a conditional for factorizing the sum of a set of gaussian integer-valued matrices
In my research project, we're exploring the decomposition of Gaussian integer-valued square matrices as a cross-product of other Gaussian integer matrices (GIM) with the same dimension. One of the ...
- 11
13
votes
1
answer
580
views
Identifying two definitions of orientation on a vector space
Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$:
A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
- 10.8k
0
votes
0
answers
87
views
Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?
Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
- 219
0
votes
1
answer
115
views
Approximation for an expectation expression
Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
CommunityBot
- 1
3
votes
1
answer
146
views
The rank of a certain linear combination of mutually commuting nilpotent matrices
Let $A_1,\ldots,A_r$ be mutually commuting $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers. For any complex number $c$, let $A(c):=A_0+cA_1+c^2A_2+\ldots +c^rA_r$. We ...
- 188
0
votes
0
answers
75
views
Orbits/affine spaces in GAP
Another GAP-related question.
I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...
- 12.9k
0
votes
0
answers
43
views
Inertia indices in GAP
Not sure that this is the right place, but I could not find a GAP specific forum.
Does anyone know if there is a built-in function in GAP to find the inertia indices of a symmetric matrix, say, over ...
- 4,959
2
votes
1
answer
183
views
Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded
Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
- 128k
2
votes
1
answer
241
views
How to solve this set of equations as efficiently as possible (with "efficiently" measured in FLOPS)?
The system of equations is the following:
$$
\Gamma_i^{\ -1} = \sum_{i=1}^nA_{ij}\Gamma_j,
$$
where $\Gamma = (\Gamma_i)$ is a vector of size $n$ and $A$ is a matrix of size $n\times n$, with $n \gt ...
- 3,065
5
votes
3
answers
1k
views
Constant rank theorem for Banach spaces
Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
- 12.9k
2
votes
1
answer
94
views
Testing for equal characteristic polynomials using a single determinant calculation
Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals.
If $p_1 \ne p_2$, then there is some positive ...
- 3,319
1
vote
1
answer
48
views
Iteration matrix representation with complex conjugate operator
I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
- 63.9k