All Questions
6,287 questions
1
vote
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63
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The rank of a matrix expression
I'm studying discrete-time LTI systems and state estimators for them. Recently, I studied this paper. I am facing a matrix rank calculation problem and having trouble solving it. I will provide more ...
- 4,484
20
votes
1
answer
557
views
Almost orthogonal maps $f:\omega \to \{-1,1\}$
Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\...
- 48.9k
4
votes
1
answer
230
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$\omega\times\omega$-Hadamard matrices
In the following, we define infinite Hadamard matrices.
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\...
- 48.9k
0
votes
0
answers
70
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Cyclotomic eigenvalue question for Distance-regular graph
I have read this paper. So, I am just thinking about if the following guess is true:
GUESS: Any Distance-regular graph (DRG) has cyclotomic character value property (which means the eigenvalues of a ...
- 109
15
votes
1
answer
649
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On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
- 178
1
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0
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58
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Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
- 11
2
votes
2
answers
227
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Is a probabilistic implementation of unitaries invertible?
Let $\{p_j\}_j$ be a set of probabilities, $\sum_j p_j = 1$, let $\{h_j\}_j$ be a set of $n \times n$ Hermitian matrices, and define $ad_h(A) $ be the adjoint.
Define the following linear mapping
$$ E(...
- 131
7
votes
3
answers
958
views
Vector of integers such that almost all dot products are positive
Let $x_1<x_2<\cdots<x_n$ be $n$ real numbers such that $\sum\limits_{j=1}^n x_j\ne0$. Do there always exist $n$ integers $a_1,a_2,\ldots,a_n$ such that
$$
\sum_{j=1}^n a_j\cdot x_j <0
\...
- 3,153
15
votes
3
answers
1k
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Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 261
3
votes
1
answer
327
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Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 43.2k
7
votes
4
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557
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Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 12.7k
2
votes
0
answers
38
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Constructing an $n$-simplex at the border of a $n$-ball by orthogonal hyperplanes
I want to construct an $n$-simplex the following way:
Choose $n$ vectors in the boundary of an $n$ dimensional ball, which are forming an $(n-1)$-simplex together.
Place the orthogonal affine $n-1$-...
3
votes
1
answer
157
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 807
2
votes
0
answers
80
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Prove uniqueness of Radon transform without using Fourier transform
The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity):
If a continuous function with compact support has zero ...
- 807
3
votes
1
answer
143
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A problem about matrix inverse and regularization methods
I'm researching the problem of solving the equation $A\mathbf{x}=\mathbf{b}$ with ill-conditioned matrices. We know that if we solve it directly, like $\mathbf{x}=\mathrm{inv}(A)\ast\mathbf{b}$, then ...
- 33
1
vote
1
answer
133
views
Graceful labeling of the complete bipartite graph and its laplacian quadratic form diagonalized
A graceful labeling of a connected simple undirected graph $G=(V,E)$ is a map $f:V\to\lbrace 1,...,|E|+1\rbrace$ such that for all $t\in\lbrace 1,...,|E|\rbrace$ there is a (trivially unique) $\langle ...
- 91
3
votes
1
answer
232
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Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
2
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1
answer
173
views
Maximizing a quadratic form involving a trace-bounded positive definite matrix?
$\newcommand{\tr}{\mathrm{tr}}$Suppose $P, Q$ are two real, symmetric positive definite matrices and $v$ a nonzero unit vector.
Consider
$$
f(X) = v^T(P + X^{-1})^{-1} v + v^T(Q + X^{-1})^{-1} v.
$$
...
- 400
6
votes
1
answer
388
views
Decimal expansion definition of real numbers, constructively
The two most common definitions of $\mathbb{R}$ are as Dedekind cuts or Cauchy sequences of rational numbers.
A real analysis student of mine is working out of the book Real Analysis and Applications ...
- 10.1k
1
vote
0
answers
40
views
learning about split cut (Integer Programming)
Here is a part of Integer Programming (Graduate Texts in Mathematics, 271) 2014th Edition.
In lemma 5.9, aiming at showing that a finite number
of splits ${(\pi, \pi_0)}$ are sufficient to generate ...
- 11
1
vote
1
answer
153
views
How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
4
votes
1
answer
170
views
About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 696
1
vote
0
answers
27
views
Seeking Help with Classifying Polygons: Waterholes and Airpockets in 2D Space
I am currently in the process of writing software and have encountered a mathematical problem. Perhaps there are some experts here who are familiar with this. It involves the classification of ...
- 11
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
1
vote
1
answer
132
views
Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
1
vote
0
answers
68
views
Low rank matrix completion with additional constraints
I have an $n \times n$ matrix $M$ of the form
$$ \sum_{i=1}^r \pi_i \frac 1 {1 - s_i} (1 - s_i)^\top,$$
where the $s_i$'s are $n \times 1$ vectors with positive entries that sum to 1, $1 - s_i$ is the ...
- 301
0
votes
1
answer
66
views
Correct conditions for the image of a matrix to intersect a cone?
Given an $m \times n$ real (or rational) matrix $A = (a_{ij})$, what are necessary and sufficient conditions for the image of this matrix to intersect a cone? I am specifically interested in the cone $...
- 11
5
votes
2
answers
189
views
Bisymmetric Hadamard matrices
Definitions: An $n\times n$ Hadamard matrix is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.
A symmetric matrix is a square matrix that is equal to its own ...
- 696
1
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0
answers
41
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Bound of entries of inverse of a unimodular matrix whose row sum is bounded
Many questions have been asked about the bound of the entries of the inverse of a matrix subject to certain conditions. Here my condition is slightly different: let $A=(a_{ij})$ be an $n \times n$ ...
- 895
10
votes
3
answers
455
views
When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
0
votes
1
answer
158
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
- 400
3
votes
1
answer
192
views
Vanishing of principal minors implies upper triangular up to permutation
Let $A$ be a square matrix. If $A$ satisfies the following two conditions
(1) $A$ is upper triangular
(2) all diagonal entries of $A$ are zero
then it is easy to see that all principal minors of $A$...
- 357
0
votes
0
answers
51
views
Minimizer of forward and reverse Kullback-Leibler divergence with sum constraints on marginals
Consider minimization of the Kullback Leibler divergence between two discrete distributions $p$ and $q$:
\begin{align*}
D_{KL} \left( p \parallel q \right) = \sum_i p_i \log \left( \frac{p_i}{q_i} \...
0
votes
0
answers
31
views
What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
0
votes
0
answers
96
views
When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
- 427
14
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0
answers
602
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
24
votes
3
answers
866
views
Mark some vectors in $\mathbb{R}^n$ in a way that every orthonormal basis has an odd number of marked vectors
Let $n$ be a natural number. Is there a set $S$ of vectors of norm $1$ in $\mathbb{R}^n$ such that every orthonormal basis of $\mathbb{R}^n$ contains an odd number of vectors from $S$?
If $n$ is odd, ...
- 679
4
votes
1
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184
views
Is there a nice basis for a pair of linear maps?
By using splitting fields I know you can put a (single) matrix in upper triangular form. This gives in my opinion the cleanest proof of the Cayley-Hamilton theorem.
consider the following...
WRONG ...
2
votes
0
answers
79
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Does every $(n-1)^2 + 1$-dimensional subspace of $n\times n$ Hermitian matrices that contains identity, contain a rank-1 matrix?
Let $M_i$, $i=1,\dots,(n-1)^2+1$, $M_1 = 1_{n\times n}$ be a set of linearly-independent Hermitian $n\times n$ matrices. Show that there exists a rank-1 matrix $P$, which is a linear combination of $...
- 628
2
votes
1
answer
326
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Full rank of Hadamard product matrix
Let $\circ$ be the Hadamard product and consider two matrices $C \in\{0,1\}^{N \times n}$ and $W\in \mathbb{R}^{N\times n}$:
$$
C:=\left[\begin{array}{cccc}
c_1^1 & c_2^1 & \cdots & c_n^1 \...
- 43
0
votes
0
answers
22
views
Eigenvalues of Composition of Hadamard Operations of Low Rank Matrices
I am interested in the eigenvalues of $$ee^T \oslash (aa^T - a^{\odot2}(a^{\odot2})^T )^{\odot \frac{1}{2}},$$ where $a \in \mathbb{R}^n$ and $e$ is the vector with all entries equal to one.
Can we ...
0
votes
0
answers
93
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
7
votes
1
answer
580
views
Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 1,759
4
votes
0
answers
284
views
Institutional approach to linear algebra
In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following
Definition. An institution ...
- 10.1k
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
2
votes
1
answer
122
views
Is it possible to solve this kind of quadratic simultaneous equations?
$$\mathbf{x} = (x_1, x_2, ..., x_N)^T \in \mathbb{R}^{N} \\
\mathbf{A}_i \in \mathbb{R}^{N \times N},
\mathbf{b}_i \in \mathbb{R}^N ,
\mathbf{c}_i \in \mathbb{R}\\
\mathbf{x}^T\mathbf{A}_i\mathbf{x}...
- 23
5
votes
1
answer
429
views
Lower bound for the rank of the sum of $n$ matrices
I found a mathematical note by George Marsaglia entitiled "Bounds for the rank of the sum of two matrices", where he proves the following result.
Let $A_1$ and $A_2$ be two complex matrices ...
- 5,215
3
votes
1
answer
99
views
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
views
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
views
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