All Questions
6,027 questions
5
votes
1
answer
474
views
An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 128k
6
votes
1
answer
239
views
Real zeroes of the determinant of a tridiagonal matrix
Let $\epsilon_1,\ldots,\epsilon_n$ be a sequence of signs and $M(t)$ be the tridiagonal matrix whose diagonal entries are $\epsilon_1 t,\ldots, \epsilon_n t$ and off-diagonal entries equal to $1$. Is ...
- 1,181
0
votes
0
answers
83
views
When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?
By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard ...
- 4,058
1
vote
0
answers
69
views
Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?
I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
- 10.5k
3
votes
0
answers
83
views
How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian.
Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting
$$f_{...
-1
votes
1
answer
77
views
Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
- 287
5
votes
0
answers
190
views
Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
3
votes
1
answer
186
views
Positive-definite block matrix with constant block sums
Given two natural numbers $n$ and $m$, suppose that $A$ is an $nm \times nm$ real nonnegative matrix. Seeing $A$ as a block matrix where each block has size $m\times m$, suppose that the sum of the ...
- 679
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 4,058
1
vote
0
answers
161
views
On an optimization question
Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
- 13.9k
0
votes
1
answer
103
views
Is it reasonable to consider the subgaussian property of the logarithm of the Gaussian pdf?
Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y_1, Y_2, \...
- 287
1
vote
0
answers
133
views
Infinite dimensional matrix solvability
In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...
- 573
4
votes
0
answers
164
views
Dimensionality reduction preserving cyclic traces
Suppose that I have $n$ matrices $A_1, \ldots, A_n \in \mathbb{R}^{m \times m}$ with $m \gg n$. Can I find $n$ new matrices $B_1, \ldots, B_n \in \mathbb{R}^{n \times n}$ that have the same 3-way ...
- 777
2
votes
1
answer
239
views
Hoeffding's Lemma for bounded complex random variables?
If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality:
\begin{align}
\mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\...
- 287
15
votes
3
answers
1k
views
Group of matrices in which every matrix is similar to unitary
$\DeclareMathOperator\GL{GL}$Let $G$ be a subgroup of $\GL_n(\mathbb{C})$ such that for every $g \in G$ there exists $c \in \GL_n(\mathbb{C})$ for which $cgc^{-1}$ is unitary (or, which is the same, $...
2
votes
0
answers
164
views
Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?
The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...
- 4,058
0
votes
0
answers
74
views
Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix
Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,058
3
votes
1
answer
67
views
Why does the normalization term disappear when computing the MLE of decomposed Gaussians
Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
- 31
1
vote
1
answer
112
views
Orthonormal matrices with columns that switch signs
Consider an orthonormal matrix $W\in\mathbb{R}^{2n\times 2n}$ that satisfies the "abs property" $$|w_i|^T |w_{i+n}|=1$$ for all $i \in \{1,2,\ldots,n\}$, where $w_i \in \mathbb{R}^{2n}$ is ...
- 13
2
votes
1
answer
433
views
Stone-Weierstrass theorem: coefficients of approximating sequence bounded?
Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$.
The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points ...
- 463
2
votes
0
answers
91
views
Ordered vector space that can be embedded into its bidual
We say that an ordered vector space $(V, \ge)$ (over $\mathbb{R}$) is "bidual embeddable" (I made up this name, not sure whether this concept already exists) if for every $x \in V$, if $x$ ...
- 346
4
votes
1
answer
141
views
Sum of holomorphic squares?
Consider a variable $z \in \mathbb{R}^n$ and assume $u(z) \in \mathbb{R}^m$ and $H(z) \in \mathbb{R}^{m \times m}$. Further assume that $H(z)$ is symmetric positive definite for every $z$. Consider ...
- 981
3
votes
1
answer
190
views
Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?
It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots.
Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
- 4,058
1
vote
1
answer
198
views
Directed graph whose adjacency matrix admits only 0 as eigenvalue
Let $G$ be a directed graph and let $P_i$
be its vertices. Let $A$
be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$
if and only if there is a directed edge from $P_i$
to $P_j$, ($a_{i,...
- 4,058
4
votes
0
answers
181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
12
votes
3
answers
784
views
Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 128k
2
votes
1
answer
112
views
The eigenvectors of adding a particular rank one matrix to the circulant matrix
Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
- 4,058
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 7,127
12
votes
1
answer
429
views
The scope of a "strong Cantor-Bernstein" property
This question is of course related to this earlier MO question, but I don't believe is answered by the posts there.
My favorite proof of the Cantor-Schroeder-Bernstein theorem actually establishes ...
- 20.5k
35
votes
0
answers
1k
views
Orthogonal vectors with entries from $\{-1,0,1\}$
Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $...
- 6,351
14
votes
1
answer
417
views
Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
- 128k
11
votes
2
answers
1k
views
Is the eigenvalue map open?
The eigenvalue map in question is
$\sigma: {\mathfrak gl}(\mathbb{C}, n) \to S_n \backslash \mathbb{C}^n$,
from $n$ by $n$ complex matrices to $\mathbb{C}^n$ vectors modulo permutation of entries by $...
- 1,123
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k
0
votes
1
answer
310
views
Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
- 13
6
votes
1
answer
300
views
Log-convexity of determinant
Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
0
votes
0
answers
146
views
Maximizing the norm of a sum of Hermitian matrices
Consider the following problem:
Problem: Given $n\times n$-Hermitian matrices $A_1,\dots,A_r$, find $e_1,\dots,e_r\in\{-1,1\}$ such that $\|e_1A_1+\dots+e_rA_r\|_\infty$ is maximized. Here the norm is ...
2
votes
1
answer
358
views
q-polynomials in terms of a basis
Consider the polynomials
$$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
3
votes
0
answers
71
views
Automorphisms of matrix algebras and Picard group
This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
- 1,766
12
votes
2
answers
1k
views
Prove/disprove a linear algebra inequality
Setting:
Suppose $\{u_i\}_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u_i^Tu_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u_i^T w> 0$ for all $i=1,\dotsc,n$. ...
- 115
6
votes
1
answer
518
views
Non-diagonalizability of the adjacency matrix of a directed graph
Let $G$ be a directed graph with no multiple edges or loops and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed ...
- 4,058
1
vote
0
answers
66
views
What is the maximum number of positive vectors such that a vector similar to only one of them can be found
I would like to know what is $k^*$ the maximum $k$ such that:
$$
\exists z_1, \ldots, z_k \in\left(\mathbb{R}^+\right)^d \quad
\forall i \leqslant k \quad \exists \hat{z}_i \in \mathbb{R}^d \quad \hat{...
- 111
3
votes
0
answers
119
views
Describing the outer automorphism of a special unitary group in terms of the Hermitian form
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
- 7,527
0
votes
0
answers
102
views
Construct a vector space whose elements are sets
I would like to construct a vector space whose elements are convex and closed subsets of $\mathbb{R}^n$.
A natural idea is as follows.
For any two sets $S_1, S_2 \subseteq \mathbb{R}^n$, define the ...
- 159
0
votes
1
answer
147
views
Seeking closed-form solution for vector equation
I'm working on a problem that involves vectors and scalar values, and I'm looking for a closed-form solution. I hope someone can help me with this or provide insights into how to approach it. Here's ...
- 111
1
vote
0
answers
223
views
Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
- 723
4
votes
0
answers
216
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
- 4,943
7
votes
1
answer
334
views
About a result on Hadamard matrix
I am currently reading "On Subsets with Cardinalities of Intersections Divisible by a Fixed Integer" by P. Frankl And A. M. Odlyzko. They used the following result without citation:
For each ...
- 281
5
votes
2
answers
721
views
Matrices with same eigenvalues
This question is a more precise version of this question.
Let's assume we have the matrix
$$\left(
\begin{array}{ccccc}
0 & a & 0 & 0 & 0 \\
f & 0 & b & 0 & 0 \\
0 &...
3
votes
0
answers
67
views
Efficiently finding solutions to the Rainbow cryptosystem using quotient spaces
Repost of a mathematics stackexchange question here as this concerns my research and it went unanswered on there.
In this paper, Ward Beullens gives another way to look at the Rainbow cryptosystem. In ...
- 173
3
votes
1
answer
309
views
Eigenvalues two-fold degenerate
Consider the matrix $$A:=\left(
\begin{array}{cccc}
0 & a & 0 & 0 \\
f & 0 & b & 0 \\
0 & e & 0 & c \\
0 & 0 & d & 0 \\
\end{array}
\right)$$
I ...