**3**

votes

**2**answers

462 views

### Linearly constrained eigenvalue problem

Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& ...

**4**

votes

**0**answers

68 views

### $SL_2(\mathbb{F})$, decomposing $\mathbb{C}\{X\}$ into irreducible $G$-representations and dimensions [on hold]

Let $\mathbb{F}$ be a finite field with $q$ elements and $H = \mathbb{F}^\times$, the multiplicative group of $\mathbb{F}$. It is known that $H$ is a cyclic group of order $q - 1$, so $\widehat{H} = ...

**4**

votes

**2**answers

48 views

### Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$.
With $nz(M) := ...

**1**

vote

**0**answers

22 views

### Uniqueness of the reduced rank QR decomposition

Let $A$ be a $n\times n$ matrix that is a real, symmetric, positive semidefinite and has rank $r$.
I read about the rank reduced $QR$ decomposition (here for example) as the QR variant where $A=QR$ ...

**3**

votes

**0**answers

45 views

+50

### How does a permutation $P$ affect the singular value $\sigma_{\text{max}}(Q^\top P^\top Q)$ for orthogonal $Q$?

Let $q_i$ for $i=1,\ldots,m$ be the columns of the matrix $Q\in\mathbb{R}^{n\times m}$, $n>m$, which are pairwise orthonormal ( i.e.
$q_i^\top q_j = \begin{cases} 1 & \text{if}\quad i=j \\ 0 ...

**4**

votes

**2**answers

215 views

### Transversality in Morse theory, linear algebra version

I am working on a product in Morse-Bott homology which has led me to the following considerations and unanswered question. I would be very grateful if anyone could help.
Suppose $H:\mathbb{R}^n \to ...

**8**

votes

**1**answer

139 views

### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here.
Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...

**2**

votes

**1**answer

252 views

### Two Concepts of Monotonicity

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that
$F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that
$$
\langle F(y)-F(x), ...

**8**

votes

**1**answer

352 views

### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...

**12**

votes

**2**answers

251 views

### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?
For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...

**2**

votes

**0**answers

61 views

### When is $\left[\begin{smallmatrix} D_1 & B \\\\ -B^T & D_2 \end{smallmatrix} \right]$ $\mathbb{R}$-diagonalizable?

Is there some block-wise characterization of $\mathbb{R}$-diagonalizability (by similarities) of
$$\begin{bmatrix} D_1 & B \\\\ -B^T & D_2 \end{bmatrix},$$
where $D_1$ and $D_2$ are real ...

**5**

votes

**1**answer

310 views

### vector spaces with uncountable dimension and a nice basis

Unlike the reals considered as a vector space over the rationals, I know of a number of nice examples of vector spaces with uncountable dimension that have a nice basis.
For example, the space of ...

**0**

votes

**1**answer

358 views

### Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.
I'm ...

**7**

votes

**0**answers

128 views

### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...

**1**

vote

**1**answer

114 views

### Sum of Eigenvectors Entries of an Adjacency Matrix

I have a question regarding the sums $\sum_{i=1}^{n}v_{j}\left(i\right)$ where $v_j$ are eigenvectors of adjacency matrix $A$ which have been normalized to unit length.
Ordering the eigenvectors by ...

**2**

votes

**2**answers

161 views

### SVD vs Fourier analysis for data.

Fourier analysis is useful for analysis in the frequency domain. SVD on the other hand is useful for analysis of data, and expressing noise in the data. I have a problem that needs extensive data ...

**1**

vote

**0**answers

39 views

### Eigenvalues of the sum of Laplacian matrix and the all ones matrix [migrated]

Given an undirected graph and its Laplacian is $L$.
I need to find the eigenvalues of the sum: $L + \mathbf{11^T}$ (where $\mathbf{1}$ is the all-ones vector, which means that $\mathbf{11^T}$ is a ...

**1**

vote

**2**answers

212 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

**43**

votes

**4**answers

5k views

### explicit big linearly independent sets

In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or ...

**7**

votes

**0**answers

64 views

### Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...

**3**

votes

**0**answers

102 views

### The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...

**8**

votes

**0**answers

189 views

### Standard polynomials applied to matrices (bis)

The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots ...

**0**

votes

**1**answer

93 views

### Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...

**0**

votes

**2**answers

120 views

### The condition number of a scaled Vandermonde matrix

Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...

**6**

votes

**3**answers

340 views

### Parameterize unitary without transpose

For all unitary matrices, i.e. $A \overline{A}^T = I$, there is a skew-Hermitian matrix $X$ so that $A = exp(X)$. So the unitary group has $n^2$ dimensions.
Is there any similar parameterisation of ...

**2**

votes

**1**answer

606 views

### What is the time complexity of approximated SVD

Full SVD, on an m*n matrix $A$, $[U,S,V] = svd(A)$, would cost $O(m^2n + mn^2 + n^3)$ time.
But what is the time complexity if we only need the $k$ largest singular values, say, $[U_k,S_k,V_k] = ...

**31**

votes

**10**answers

6k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**15**

votes

**5**answers

3k views

### Is there a version of inclusion/exclusion for vector spaces?

I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...

**3**

votes

**0**answers

76 views

### Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the ...

**8**

votes

**2**answers

153 views

### generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix
the maps
$$
f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$
x\longmapsto ...

**1**

vote

**1**answer

101 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 ...

**1**

vote

**0**answers

87 views

### Notions of singularity for symmetric bilinear maps

Given a symmetric, bilinear map $B : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$, a common notion of nondegeneracy is the following:
$$
\mbox{If } B(x,y) = 0 \mbox{ for all } y, \mbox{then } ...

**19**

votes

**1**answer

639 views

### Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here.
Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Lemma 2. For any pair $i$, $j$ such that $0 ...

**2**

votes

**2**answers

143 views

### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...

**6**

votes

**1**answer

221 views

### How to calculate expected value of matrix norms of $A^TA$?

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with ...

**3**

votes

**1**answer

146 views

### Smallest eigenvalue gap of a non-symmetric random matrix

The question: Let $A$ be the matrix whose each element is an independently generated random variable which is uniform on $[0,1]$. One can see that the eigenvalues of $A$ will be distinct almost ...

**3**

votes

**1**answer

145 views

### Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$,
we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$
and $T$ be sets of elements of $\mathbb{F}$ of size ...

**3**

votes

**1**answer

228 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**8**

votes

**1**answer

180 views

### Pair of square matrices related by traces formulas

Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{C}$. Assume that for every $k\geq 1$ it holds $tr(A^k) = tr(B^{2k-1})$. What can we say about the possible eigenvalues of $A$ and of $B$? How ...

**3**

votes

**1**answer

120 views

### comparing norms of tensor product of two Hilbert spaces

Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...

**2**

votes

**1**answer

131 views

### Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form
$$\begin{bmatrix}
\pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\
A_{2n} & ...

**-1**

votes

**1**answer

32 views

### finding a unitary submatrix inside a random matrix

Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...

**1**

vote

**1**answer

49 views

### Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k} $ with $n \leq k $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions:
...

**2**

votes

**0**answers

25 views

### Nonconvex function on the singular value vectors of projected matrix

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...

**0**

votes

**0**answers

33 views

### Triangle inequality for nonconvex functions of singular value vector

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...

**12**

votes

**1**answer

334 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**18**

votes

**2**answers

745 views

### a determinantal identity

Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity
$$
\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)
$$
...

**5**

votes

**1**answer

257 views

### Can I find the gap between the two least eigenvalues of this special matrix A(t)?

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal ...

**18**

votes

**2**answers

2k views

### Dimension of infinite product of vector spaces

This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...

**2**

votes

**1**answer

81 views

### Convergence of Symmetric Iterative Proportional Fitting

Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by ...