# Tagged Questions

**2**

votes

**1**answer

78 views

### How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?

Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...

**6**

votes

**2**answers

281 views

### Parametrization of positive semidefinite matrices

We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...

**1**

vote

**1**answer

244 views

### Decompose $\Lambda^3(V \otimes W)$ [closed]

Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes ...

**4**

votes

**0**answers

76 views

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

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} = ...

**3**

votes

**1**answer

70 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 $A=QR$ ...

**4**

votes

**2**answers

54 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) := ...

**12**

votes

**0**answers

176 views

### 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 $Q\in\mathbb{R}^{n\times m}$, $n\geq 2m$, which are pairwise orthonormal ( i.e.
$q_i^\top q_j = \begin{cases} 1 & \text{if}\quad i=j \\ 0 ...

**12**

votes

**2**answers

288 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

71 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 ...

**8**

votes

**0**answers

139 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

139 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 ...

**8**

votes

**0**answers

150 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 ...

**1**

vote

**2**answers

220 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 ...

**8**

votes

**0**answers

73 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}+ ...

**4**

votes

**0**answers

122 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 ...

**3**

votes

**0**answers

79 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

166 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

**0**answers

90 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 } ...

**20**

votes

**1**answer

679 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 ...

**7**

votes

**1**answer

274 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

158 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 ...

**8**

votes

**1**answer

186 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

134 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 ...

**-1**

votes

**1**answer

39 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

103 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

34 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 ...

**0**

votes

**1**answer

47 views

### When to use non-negative-least square and least-square [closed]

What are the typical case we need to use Non-negative least squares NNLS
$$
||Ax - B||^2
$$
instead of least-square $$ Ax-B$$ (or vice versa)?
And is there any drawback in applying them on large ...

**5**

votes

**1**answer

101 views

### Maps between products of symmetric powers

This question might be too elementary but it arises naturally as a part of a more complicated computation and I struggle to find the answer.
Let $V$ be an $n$-dimensional complex vector space. ...

**1**

vote

**2**answers

101 views

### Irreducibility of tuples of matrices

I am interested in tuples of real or complex $d\times d$ matrices which are irreducible in the sense that their constituent matrices do not admit a common invariant linear subspace whose dimension is ...

**10**

votes

**3**answers

294 views

### Finite sets of vectors closed under an orthogonality property

Say that a set $X \subset \mathbf{S}^{d-1}$ is ortho-closed if for any set $\{x_1,\dots,x_{d-1}\} \subset X$ there exists $x \in X$ such that $\langle x,x_i \rangle = 0$ for $i=1,\dots,d-1$. ...

**5**

votes

**1**answer

106 views

### What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix ...

**0**

votes

**0**answers

29 views

### On the computation of generalized eigenvalues of a low-rank approximation using SVD

I have trouble deriving an expression of the generalized eigenvalues of a matrix pair, found in http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4618700 .
The setup is the following and can ...

**2**

votes

**0**answers

50 views

### Partial inverse of a matrix exponential

The partial inverse of a matrix, also known as principal pivot transform among other names, is defined as follows. if
$$
M\begin{pmatrix}x_a\\x_b\end{pmatrix} = \begin{pmatrix}y_a\\y_b\end{pmatrix}
$$
...

**9**

votes

**2**answers

279 views

### Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?

When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...

**2**

votes

**1**answer

187 views

### Is this affine-subspace analogue of a Grassmannian a classifying space?

Let $AG_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG_k(\mathbb{R}^N)$ and the ...

**6**

votes

**2**answers

357 views

### Counting matrices over finite fields of a given order

How can I count/enumerate matrices in ${\rm GL}(2,{\rm GF}(2^5))$
of order $3$? In general, how can I obtain the number of matrices in
${\rm GL}(2,{\rm GF}(q))$, where $q$ is a power of a prime, of ...

**16**

votes

**4**answers

909 views

### Reference for a linear algebra result

I asked the following question (http://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no ...

**6**

votes

**3**answers

346 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 ...

**3**

votes

**2**answers

112 views

### Example for Reciprocal Principal Minors

I'm searching for rather specific counter-example.
Some notation: $A(\alpha|\beta)$ is the sub matrix of $A$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: ...

**5**

votes

**1**answer

168 views

### Is this subset-sum-type problem discussed in the literature?

Let $y \in \mathbb{Z}_+^n$, with $y_1 < \dots < y_n$. I am interested in finding a 0-1 matrix $A$ and $x \in \mathbb{Z}_+^m$ s.t. $m$ is minimal and $Ax = y$, where I am guaranteed that at least ...

**3**

votes

**0**answers

89 views

### What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?

For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...

**7**

votes

**0**answers

110 views

### A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$

In the appendix of the paper by Tolhuizen (
http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...

**1**

vote

**0**answers

91 views

### For of a special case of $Ax>=b$, are there always integer solutions?

The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$.
The output is a matrix $A(n\times n)$ ...

**1**

vote

**0**answers

41 views

### Product of elementary divisors

Let $A$ be an $(m \times n)$ integer matrix (if it helps, we can assume that a is a square matrix). Let $d_i,\ldots,d_s$ be the elementary divisors of $A$. I am interested in the product ...

**5**

votes

**1**answer

73 views

### Is there a class of functions acting on a set of projected points that remain invariant under changes in projection parameters?

Suppose I have a set of $k$ points $\{x_1,x_2,\ldots,x_k\}$ in $\mathbb{R}^n$ that I can project into $\mathbb{R}^m$ with the linear operator $\mathcal{P}$, with $\alpha, \beta, \ldots$ parameters of ...

**6**

votes

**1**answer

104 views

### Invertible combinations of linear maps on infinite-dimensional vector spaces

Let $V$ be a real infinite-dimensional vector space of cardinality $\kappa$. Does there exist a set $\Omega$ of cardinality $\kappa$ of linear maps from $V$ to $V$ such that for every $n\geq 1$, every ...

**4**

votes

**0**answers

64 views

### How does scaling rows to sum to 1, of a positive matrix change the perron vector?

Reposting from math.sx due to lack of response. Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron ...

**3**

votes

**1**answer

98 views

### Orbits in the adjoint representation of $SU(2,1)$

How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?

**2**

votes

**0**answers

55 views

### The characteristic polynomial of the product of two linear recurrences

Let $\mathbb{F}$ be a field and let $(a_n)_{n \geq 0}$, $(b_n)_{n \geq 0}$ be two linear recurrences with terms in $\mathbb{F}$ and respective characteristic polynomials $f(X), g(X) \in ...