# Tagged Questions

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12 views

### Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil

Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...

**3**

votes

**1**answer

55 views

### Homogeneous polynomial vector fields tangent to the unit sphere

This question has something to do with that one.
Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v_1,\ldots,v_n)$ whose components $v_j$ are homogeneous of ...

**2**

votes

**1**answer

126 views

### An expectation of the product of random unitaries

I want to find the answer of
$$\int dU \ U^m X \ U^{\dagger m}$$
Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...

**5**

votes

**2**answers

222 views

### Expectation of trace of nth power of unitary matrices

I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...

**0**

votes

**0**answers

23 views

### Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$
where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is
$$
\Lambda = \begin{bmatrix}
x ...

**1**

vote

**0**answers

48 views

### Are A and A^T unitarily equivalent over a p-adic field?

Let $E/F$ be a quadratic extension of p-adic field. Let $U=\{u\in GL_2(E): uu^*=1\}$ be the unitary group of rank 2.
My question is: given a matrix $A\in GL_2(E)$ can we find $u_1,u_2\in U$ such ...

**0**

votes

**0**answers

27 views

### About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing.
Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph.
I would like to understand what is the ...

**1**

vote

**1**answer

166 views

### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log_2n)^{\frac{1}{c+1}})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
...

**4**

votes

**1**answer

128 views

### Can every finite abelian $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers?

Let $G$ be a finite abelian $p$-group (where $p$ is a prime). Suppose there exists a symmetric bilinear map $\delta\colon G\times G\to \mathbb{Q}/\mathbb{Z}$ such that the induced map $g\to\langle ...

**1**

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**0**answers

22 views

### Phase of the inner product between the elements of an ETF

I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question:
Is there any research about the phase of inner ...

**5**

votes

**1**answer

389 views

### Some calculus in the orthogonal group $O(n)$

How can one compute each of the following matrices, explicitly:
$$\int_{O(n)} e^{g}dg$$ or
$$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$
What is the explicite entries of the resulting ...

**1**

vote

**0**answers

55 views

### Relations in a space generated by indicator functions

Simple Question
I ran into the following seemingly simple question.
For an arbitrary set $M$ consider the real vector space generated by indicator functions $\chi_A$ of all subsets $A\subset M$. ...

**7**

votes

**0**answers

78 views

### Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...

**0**

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**0**answers

199 views

### “Bridging the gap” to research level linear algebra [closed]

I would like to ask if there exist reference books to "bridge the gap" between the material presented in Lang's Linear Algebra and current research in linear algebra. Put another way, what is the ...

**1**

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**0**answers

173 views

+50

### A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...

**0**

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**0**answers

16 views

### Column Subset Selection implementations

Are there readily available implementations of algorithms for the CSSP - Column Subset Selection Problem?

**2**

votes

**1**answer

83 views

### Normalizing a matrix via triangular transformations

Consider the following $n\times n$ real-valued matrix:
$$
A=\begin{bmatrix}
\alpha_1 & \beta_1 & 0 &\cdots & 0 &\gamma_n\\
\gamma_1 & \alpha_2 & \beta_2 & \cdots & ...

**4**

votes

**2**answers

195 views

### An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...

**3**

votes

**1**answer

172 views

### Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...

**2**

votes

**0**answers

41 views

### Computing maximal ideals of a Lie algebra

Would you know an algorithm (or an automatic method) that computes all maximal ideals $J$ of a given Lie algebra?
Or an algorithm that computes all maximal ideals $J$ containing a given minimal ideal ...

**5**

votes

**1**answer

275 views

### Examples of Polyhedra with Large Shadows

Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...

**0**

votes

**1**answer

74 views

### What is exponentially fitted osculating straight line?

While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...

**0**

votes

**1**answer

83 views

### When is a $2$-lift of a graph connected? [on hold]

Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...

**1**

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**0**answers

46 views

### Lower bound on difference between polynomials at moderate distance

Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i ...

**2**

votes

**0**answers

126 views

### Is there a symmetric basis for $\mathbf{Q}(x,y)$?

Consider $\mathbf{Q}(x,y)$, the rational functions in $x$ and $y$, as a vector space over $\mathbf{Q}$.
Let $\sigma$ be the map interchanging $x$ and $y$. Is there a basis for $\mathbf{Q}(x,y)$ ...

**0**

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**0**answers

57 views

### Signed Laplacians and Ramanujan graphs

Given a signing/2-lift matrix $A_s$ of a $d-$regular graph one has the relationship that the ``Signed Laplacian" is $L = d + A_s$. This $L$ is still the same size as the base graph. But the lifted ...

**1**

vote

**1**answer

51 views

### Laplacian spectrum of $2-$lifts of graphs

We know that a $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a signing matrix) denoting which edge is to be duplicated by the identity permutation on ...

**3**

votes

**1**answer

113 views

### How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...

**2**

votes

**2**answers

112 views

### Positive solutions of linear systems with a diagonally dominant matrix

Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^*$ considering that $\mathbf{A}$ is diagonally dominant? ...

**2**

votes

**0**answers

103 views

### Number of degree $k$ functions [closed]

Given a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, there is a real multivariate multilinear polynomial that is associated with in through interpolation.
Example: ...

**0**

votes

**0**answers

48 views

### Invariant subspace of bounded self-adjoint operator [migrated]

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum.
Now, I was ...

**0**

votes

**2**answers

68 views

### Simple Spectrum of Jacobi matrices

I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...

**4**

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**0**answers

441 views

### Can you equip every vector space with a Hilbert space structure? [migrated]

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$.
Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...

**0**

votes

**1**answer

125 views

### Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are ...

**0**

votes

**2**answers

62 views

### About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...

**1**

vote

**0**answers

29 views

### Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...

**5**

votes

**0**answers

178 views

### Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia):
$$
\lambda^\downarrow(A) \cdot ...

**11**

votes

**5**answers

754 views

### Does k(X) have a k-basis for every set X, without AC?

This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?.
For any field $k$, the field $k(x)$ of rational functions in one variable has an ...

**1**

vote

**1**answer

133 views

### Are there good ways of relating a minor to the full determinant?

Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...

**-1**

votes

**1**answer

70 views

### How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable?
In this context PSD means symmetric PSD.

**1**

vote

**0**answers

54 views

### Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model
\begin{equation}
\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}
\end{equation}
where ...

**2**

votes

**0**answers

65 views

### Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?

Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.)
I want to calculate or bound $\det ...

**0**

votes

**0**answers

51 views

### Shift functor and the origin of the linear decalage isomorphism

Let $(\mathbf{Vec}_\mathbb{Z}(\mathbb{K}),\otimes,\tau)$ be the symmetric monoidal category of $\mathbb{Z}$-graded $\mathbb{K}$-vector spaces, where $\otimes$ is the tensor product of ...

**8**

votes

**0**answers

114 views

### Intersection of Springer fibre and Schubert cell

Let us consider intersections of Springer fibres and Schubert cells in type A.
Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let
$$
F_Y = \{ V_0 = 0 \subset V_1 \subset ...

**0**

votes

**0**answers

85 views

### Question about majorization of eigenvalues after conjugation

Let $A$ and $B$ be $n \times n$ positive semidefinite matrices with eigenvalues $\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$ respectively. ...

**0**

votes

**0**answers

66 views

### The norm of a Finite Hilbert matrix

Let $H$ be an $n\times n$ Hilbert matrix,
$$h_{ij}=(i+j-1)^{-1}.$$
The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left ...

**12**

votes

**3**answers

826 views

### How do we show this matrix has full rank?

I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988):
Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
...

**1**

vote

**0**answers

115 views

### Distinct determinants of circulants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in?
For $n=1,2,3,4$, I ...

**3**

votes

**0**answers

95 views

### Dimension of certain polynomial spaces

Let $(\omega_1, \eta_1） \dots (\omega_n, \eta_n)$ be $n$ pairs of complex numbers where $\omega_i \ne \omega_j$ for all $1 \leq i \ne j \leq n$. We define the following polynomial space
$$
Z_n^d(\eta, ...

**3**

votes

**1**answer

278 views

### Open problems in compressed sensing

What are the main open problems in compressed sensing?
I am interested in theoretical as well as in numerical point of view.