All Questions
6,027 questions
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
3
votes
1
answer
153
views
Solving a recursion for polynomials defined by a matrix product
Define the polynomial $p_n(X) \in \mathbb{Z}[X_1,...,X_n]$ as the top left entry in $A^n$ for the $(d \times d)$ matrix
\begin{align*}
& A = \left(\begin{matrix}
X_1 & \dots & \...
- 1,295
4
votes
0
answers
108
views
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
2
votes
1
answer
99
views
Stabilizing conjugacy classes of integer matrices
$\DeclareMathOperator{\Conj}{Conj} \DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\id}{id} \newcommand\Z{\mathbb{Z}}$
For an $n \times n$ integer matrix $A \in \GL_n(\Z)$, let $\Conj(A)$
be the ...
- 23
7
votes
2
answers
347
views
Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...
- 541
2
votes
1
answer
278
views
Continuity of eigenvector of zero eigenvalue
Wonder whether anyone has an idea on showing the following or to point out that it is not true:
Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
- 69
2
votes
2
answers
215
views
How to analyze the value of convergence of functions of random matrices?
Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
- 287
0
votes
1
answer
222
views
Low rank matrix in a subspace of matrices
Let $V\subset M_{m,n}(\mathbb{C})$ be a $n$-dimensional subspace ($m\geq 2$).
Is there $X\in V$ such that $1\leq \operatorname{rank} X\leq m-1$?
- 13
0
votes
1
answer
104
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 ...
- 887
5
votes
1
answer
489
views
Does coefficient-wise limit preserve real-rootedness?
Let $P_n$, $n=1,2,\ldots$ be polynomials with real roots only (and real coefficients), and $P_n$ converge to a non-zero polynomial $Q$ coefficient-wise. Does it follow that $Q$ has real roots only?
...
- 109k
0
votes
1
answer
60
views
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: $...
- 455
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 $R^n$, ...
- 33
0
votes
1
answer
525
views
What is the mathematician's definition of the determinant? [closed]
I am trying really hard to find a good definition of the determinant.
I have looked virtually every single resource online and everybody gives a different answer:
sum of cofactors or minors https://...
- 179
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 ...
- 37.7k
1
vote
1
answer
141
views
Does a random matrix over $\mathbb{Z}_q$ map linearly independent vectors to statistically independent vectors?
Suppose $A \in \mathbb{Z}_q^{n \times m}$ is a random $n \times m$ matrix whose entries are i.i.d. uniform over $\mathbb{Z}_q=\mathbb{Z}/q\mathbb{Z}$, where $q\geq2$. Let $\mathbf{x}_1, \ldots, \...
- 503
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 ...
- 503
6
votes
2
answers
623
views
Can this system of equations about Newton's formula have concrete result?
Try to solve this system of equations:
$$
S_1=x_1+\dots+x_n=a;\\
S_2=x_1^2+\dots+x_n^2=a;\\
{}\cdots\\
S_n=x_1^n+\dots+x_n^n=a;
$$
And find the value of $S_{n+1}=x_1^{n+1}+\dots+x_n^{n+1},a\in\mathbb{...
- 75
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
34
votes
3
answers
6k
views
Why is uncomputability of the spectral decomposition not a problem?
Below, we compute with exact real numbers using a realistic / conservative model of computability like Type Two Effectivity.
Assume that there is an algorithm that, given a symmetric real matrix $M$, ...
- 4,943
1
vote
0
answers
255
views
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 ...
- 275
3
votes
1
answer
317
views
"Totally real" linear transformations
Identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ via the equality $$(z_1, z_2, \ldots, z_n)=(x_1, \ldots, x_n, y_1, \ldots , y_n)$$
Where $z_j=x_j + iy_j$.
We call a linear invertible map $A: \mathbb{R}^...
user515519
1
vote
1
answer
1k
views
Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...
2
votes
1
answer
115
views
Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane
Suppose that
$f$ and $g$ are polynomials with nonnegative coefficients,
the degree of $g$ is greater than the degree of $f$,
$g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
- 187
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
1
vote
2
answers
318
views
Right inverse of integer matrix
If I have a rectangular matrix $A$ (say $4 \times 6$) with integer entries, is there a way to tell whether it has a right inverse that also has integer entries. I know that if $AA^T$ has determinant $...
- 71
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 ...
1
vote
1
answer
345
views
Solvability of $A X B=C$ with $X=X^\mathrm{T}$
I am studying symmetric solutions to the complex matrix equation
\begin{equation}
A X B=C,
\end{equation}
where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
- 61
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
5
votes
1
answer
186
views
What is expected (border) rank of the knonecker product of 3-tensors
Given two three order tensors $T$ and $S$ in $F^{m\times n\times p}$ and $F^{a\times b\times c}$. Clearly $\operatorname{rk}(T\otimes S)\le \operatorname{rk}(T)\operatorname{rk}(S)$. Does the equality ...
- 151
1
vote
1
answer
206
views
Reflections on subspaces of $\text{codim} > 1$
Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
- 1,813
0
votes
0
answers
72
views
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
34
votes
3
answers
3k
views
Quickly determining if a matrix has any PSD completion
Given $m$ entries of an $n \times n$ matrix, is it possible to determine in $O(m n)$ time whether there is any positive semidefinite completion?
Slightly more precisely: for simplicity let's assume ...
- 777
1
vote
1
answer
178
views
Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant
Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
- 3,741
6
votes
1
answer
427
views
When are the chirp signals orthogonal?
Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
- 287
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 ...
0
votes
0
answers
121
views
Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
- 1,305
7
votes
1
answer
307
views
Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric
Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
- 173
0
votes
0
answers
184
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
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
11
votes
6
answers
872
views
A question on the real root of a polynomial
For $n\geq 1$, given a polynomial
\begin{equation*}
\begin{aligned}
f(x)=&\frac{2+(x+3)\sqrt{-x}}{2(x+4)}(\sqrt{-x})^n+\frac{2-(x+3)\sqrt{-x}}{2(x+4)}(-\sqrt{-x})^n \\
&+\frac{x+2+\...
- 145
1
vote
0
answers
130
views
A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
10
votes
2
answers
895
views
What norms can be "universally" defined on any real vector space with a fixed basis?
Let $V$ be a real vector space and let $B = (b_\lambda)_{\lambda \in \Lambda}$ be a basis. So every $v \in V$ can be written uniquely as a linear combination
$$ v = c_{\lambda_1} b_{\lambda_1} + c_{\...
- 641
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
15
votes
1
answer
2k
views
Real polynomials that go to infinity in all directions: how fast do they grow?
Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
- 26.9k
0
votes
0
answers
66
views
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
173
views
Show that $\mathrm{PSL}_2(C)$ is complex algebraic [closed]
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\im{im}$I meet this problem when reading Artin's book Algebra. ...
- 225
4
votes
0
answers
1k
views
Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
- 47
2
votes
1
answer
213
views
Is matrix B obtained from matrix A?
Assuming a matrix $\mathbf{A} \in \mathbb{R}^{4096 \times 4096}$ sampled from a standard normal distribution $N(0, 1)$, and another matrix $\mathbf{B} \in \mathbb{R}^{4096 \times 4096}$ either sampled ...
- 49
3
votes
1
answer
132
views
Existence of a density
Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite ...
- 301
2
votes
1
answer
358
views
q-polynomials in terms of a basis
Consider the polynomials
$$f_n(q)=\prod_{j=1}^n(1+q^j) \qquad \text{and} \qquad g_m(q)=1+q+q^2+\cdots+q^m.$$
I'll list a few examples to motivate my question. Direct calculations show that
$$f_1=g_1, \...
- 43.2k