**7**

votes

**1**answer

767 views

### A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...

**5**

votes

**3**answers

540 views

### Set of Positive Definite matrices with determinant > 1 forms a convex set

While reading a paper An Arithmetic Proof of John’s Ellipsoid Theorem by Gruber and Schuster, I have a question on their proof.
Consider an $n\times n$ real symmetric and positive definite matrix ...

**2**

votes

**1**answer

79 views

### Updating $LU$ decomposition after adding a sparse matrix

How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing ...

**4**

votes

**2**answers

228 views

### Diagonalization of quaternion matrices

I would like to diagonalize a very large matrix that has the same property as quaternion matrices (in a sense that the matrix can be written as a linear combination of quaternions):
$$
H = ...

**2**

votes

**0**answers

100 views

### A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...

**0**

votes

**0**answers

207 views

### Matrix inversion lemma and pseudoinverse?

Starting with this form of the matrix inversion lemma
$$
(A^T B^{-1} A + D^{-1})^{-1} A^T B^{-1} = D A^T (B + A D A^T)^{-1}
$$
substitute $B \rightarrow I$, $D \rightarrow \alpha^{-1} I$,
$$
...

**4**

votes

**3**answers

202 views

### Sherman-Morrison type formula for Moore-Penrose Pseudoinverse

Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$.
Then the ...

**0**

votes

**0**answers

192 views

### expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...

**5**

votes

**3**answers

290 views

### Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R ...

**2**

votes

**1**answer

82 views

### Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...

**2**

votes

**0**answers

105 views

### Kullback-Leibler Divergence of Stationary Distributions of Markov chains

Consider two finite Markov chains on the same state space, both assumed to be irreducible, with transition matrices $P$ and $Q$ and associated stationary distributions $\pi$ and $\tilde \pi$. Is it ...

**0**

votes

**1**answer

64 views

### Is the Hodge Map Unitary?

Let $(V,< \cdot, \cdot >)$ be an inner product space over a field ${K}$. As usual, we can extend $< \cdot, \cdot >$ to a mapping on the exterior algebra of $V$ using the usual matrix ...

**4**

votes

**0**answers

187 views

### Probability distribution function for singular value sum of Gaussian random matrix

Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition ...

**0**

votes

**0**answers

60 views

### Constructing a digraph from its spectrum

This is related to the following question from cs theory stack exchange:
http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem
So it seems as if given a sequence of real ...

**5**

votes

**1**answer

202 views

### Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated.
Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each
...

**2**

votes

**0**answers

134 views

### Expected mean square error of an estimation problem

Let R be an $(N+1)\times (N+1)$ Toeplitz matrix. I would be mostly interested in the case of $N\to \infty$. Let $x$ be a complex Gaussian random $N\times 1$ vector with mean zero and covariance matrix ...

**3**

votes

**1**answer

135 views

### Maps between general linear group that can be extended to functor

this is just a basic linear algebra question, which I do not have a idea.
Suppose that we have a group homomorphism $\phi:GL_n(\mathbb{R})\rightarrow GL_m(\mathbb{R})$. Is there always a functor ...

**2**

votes

**0**answers

120 views

### equalizing diagonals of a matrix inverse

Let $Z$ be a square matrix; denote by $\text{diag}_k(Z)$ the matrix containing the center $2k+1$ diagonals of $Z$; i.e., if $T=\text{diag}_k(Z)$, then
$$T_{ij}=Z_{ij}, \;\mathrm{if}\; |i-j|\leq k, ...

**6**

votes

**2**answers

218 views

### Powers of singular matrices and pairs of identical rows

Let $A$ be a square real or complex matrix. We’ll call $A$ special if among its rows (or among its columns) there are two identical ones, different from the zero vector, (Added:) and if it has no ...

**11**

votes

**2**answers

401 views

### How to project a vector onto a very large, non-orthogonal subspace

I have a difficult problem.
I have a very large, non-orthogonal matrix $A$ and need to project the vector $y$ onto the subspace spanning the columns of $A$. If this were a small matrix, I would use ...

**2**

votes

**3**answers

530 views

### Are all (possibly infinite dimensional) irreducible representations of a commutative algebra one-dimensional?

If $A$ is a commutative algebra over an algebraically closed field $k$, and $\rho:A \rightarrow End(V)$ is an irreducible representation of $A$ (where, a priori, $V$ may be infinite dimensional), can ...

**4**

votes

**1**answer

217 views

### The Maslov triple product is alternating in its entries

Let $(V,\omega)$ be a $2g$-dimensional symplectic vector space. I'm trying to understand the Maslov triple product. I know that it can be defined in a variety of ways, but for the applications I'm ...

**5**

votes

**1**answer

304 views

### Can one characterize the category of finite-dimensional vector spaces? [duplicate]

Let $K$ be a field. Does the category of finitely generated $K$-modules have a nice characterization, for example as the unique abelian category satisfying a certain simple condition? For example, we ...

**4**

votes

**0**answers

68 views

### Level sets of linear combinations of Gaussians

I am trying to work out whether level sets of linear combinations of Gaussian functions are unique.
For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let ...

**4**

votes

**1**answer

105 views

### Distribution of the spectrum of a perturbed matrix

Let $A$ be an $n\times n$ Hermitian matrix,
with well-separated eigenvalues $\lambda_1 > \lambda_2 ... > \lambda_n$,
with $|\lambda_i-\lambda_j|>\epsilon$, for all $i \neq j$.
Let $G$ be a ...

**2**

votes

**2**answers

112 views

### Given a subdomain of GL(n), when is the map from matrices to their matrices of eigenvectors a diffeomorphism?

I'm wondering if there are any general conditions on a subdomain of $GL(n)$, which would guarantee that the map from a matrix to its matrix of eigenvectors is a diffeomorphism.
For example, given a ...

**8**

votes

**3**answers

1k views

### How to solve this quadratic matrix equation?

I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...

**6**

votes

**4**answers

462 views

### A name for matrices with only simple eigenvalues?

I am constantly working with hermitian matrices without multiplicity in their spectrum. Since this hypothesis appear in several important problems, for instance perturbation theory, I looked in the ...

**2**

votes

**1**answer

155 views

### positive semidefinite matrix condition

There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...

**4**

votes

**0**answers

100 views

### Concept of eigenvector restricted to nonnegative entries

Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem
$\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...

**2**

votes

**1**answer

110 views

### How to solve a matrix equation with both inverses and a hadamard product?

I have a matrix equation of the form:
$$
A^{-1} = B + A \circ C
$$
where $\circ$ denotes the Hadamard product (i.e., $(A\circ C)_{ij} = A_{ij}B_{ij}$). How can I determine if a solution for $A$ ...

**3**

votes

**1**answer

156 views

### Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem
\begin{align}
&\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\
...

**3**

votes

**1**answer

119 views

### Condition number of a random 0-1 matrix

Consider a 0-1 integer $n \times n$ matrix with coefficients chosen uniformly over $\{0,1\}$. The probability that it is singular is exponentially small, and so we expect that it has a well-defined ...

**3**

votes

**1**answer

117 views

### Is there an algorithm to compute group presentations of or find generators for the centralizer of a matrix in $GL(n, \mathbb{Z})$?

Let $M \in H \leq GL(n, \mathbb{Z})$. Is there an algorithm that computes either matrix generators or even a group presentation for $C_H(M)$ given generators or a presentation of $H$? Also is ...

**2**

votes

**1**answer

138 views

### When is there a solution to these coupled eigenvalue equations?

I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...

**5**

votes

**1**answer

224 views

### Does this cross-product norm inequality hold?

I asked this on MSE over a month ago, but the one answer I got doesn't seem to work.
Let $\times$ denote the cross-product. $\;$ Is it the case that
For all unit vectors $\:\mathbf{x}\hspace{.01 ...

**14**

votes

**4**answers

792 views

### Condition for two matrices to share at least one eigenvector?

Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so ...

**5**

votes

**2**answers

356 views

### Conjugacy classes of PGL(3,Z)

We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$.
I am interested in ...

**2**

votes

**3**answers

231 views

### eigenvalue of Laplacian matrix

If we have a Laplacian matrix $\boldsymbol{A}$ such that
\begin{align}
&A_{ii} >0 \\
&A_{ii}=-\sum_{j\neq i}A_{ij}
\end{align}
with known eigenvalues $\lambda_i$.
Define the matrix ...

**2**

votes

**1**answer

105 views

### Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]

Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 ...

**7**

votes

**2**answers

165 views

### Sum of Difference of anti-diagonal matrix elements

Let $A \in \mathbb{R}^{n \times n}$, with elements $a_{ij}$
What conditions on $A$ are required for the following to be true?
There exists some vector $x \in \mathbb{R}^n_+$, $x \neq 0$ such that ...

**23**

votes

**6**answers

1k views

### Does seeing beyond the course you teach matter? The case of linear algebra and matrices

This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...

**3**

votes

**1**answer

118 views

### On matrices conjugated in a faithful representation

Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...

**0**

votes

**0**answers

163 views

### Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims:
Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...

**2**

votes

**0**answers

68 views

### The minor of a square matrix

Given a $n \times n$ matrix, the $(i, j)$ minor is the determinant of the submatrix formed by deleting the i-th row and j-th column. If the sum of all row vectors and the sum of all column vectors are ...

**0**

votes

**0**answers

88 views

### Operator Adjoints and Non-Symmetric Inner Products

Let $V$ be a finite dimensional vector space (over $C$ if that makes a difference), and let $T$ be a linear operator on $V$. Now if $(\cdot,\cdot)$ is an inner product on $V$, then it is well-known ...

**2**

votes

**0**answers

128 views

### Invariant subspaces of permutation matrix [closed]

Let $\sigma$ be a permutation matrix of order $n$. What are all the invariant subspaces of $\sigma$?
(I can only find 1 and n-1 dimensional subspaces)
Thanks in advance.

**0**

votes

**1**answer

141 views

### Bounding the positive semi-definite matrix with its block diagonal matrix [closed]

Can we bound $\mathbf{A}$ with $\mathbf{A^*}$ as ${\bf{A}} \preceq {{\bf{A}}^*}$ where
\begin{equation}
{\bf{A}} = \left[ {\begin{array}{*{20}{c}}
{{{\bf{A}}_{11}}}&{...}&{{{\bf{A}}_{1N}}}\\
...

**5**

votes

**0**answers

191 views

### Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity

Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for ...

**3**

votes

**2**answers

203 views

### Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$

I'm using the following result in a computer science paper:
Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let
$$V^\perp = \{u \in ...