**0**

votes

**1**answer

43 views

### Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices.
Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...

**0**

votes

**0**answers

7 views

### dual basis of cohomology algebra [migrated]

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients.
Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$.
Let the dual basis be
...

**1**

vote

**0**answers

47 views

### Notions of consistency / heterogeneity in sets of vector values?

The problem
Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1):
$$u=(u_1 \cdots u_n), n\in\mathbb{N}$$
$$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$
I ...

**1**

vote

**1**answer

89 views

### Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.

**0**

votes

**0**answers

60 views

### Hadamard / matrix product adjoint

First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself).
I apologize if this question seems dumb; I'm a new ...

**1**

vote

**0**answers

67 views

### Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...

**6**

votes

**5**answers

270 views

### The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.
Is there any ...

**1**

vote

**0**answers

82 views

### Showing a wedge product is nonzero

Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for ...

**1**

vote

**1**answer

71 views

### Expectation of Gaussian random vector & arbitrary function thereof?

I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity:
where the <.> operator refers to a population average.
No source or ...

**5**

votes

**0**answers

202 views

### Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...

**-3**

votes

**0**answers

47 views

### On ranks of matrix products [closed]

Tensor product of two matrices increases simultaneously sizes of product matrix, size of rank multiplicatively.
Is there a function on two matrices which increases size multiplicatively while rank ...

**4**

votes

**0**answers

71 views

### Geometric interpretation for partial trace?

This MO question asks for a geometric interpretation of the trace of a linear transformation. I'm wondering about a geometric interpretation of partial trace.
Given a linear transformation $f: ...

**2**

votes

**1**answer

157 views

### Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...

**0**

votes

**0**answers

169 views

### Theorems about matrices with entries from $0,1,-1$? [closed]

Consider matrices which are of the form $\left [ \begin{matrix}
0 && A\\
A^T && 0 \\
...

**0**

votes

**0**answers

85 views

### About distinct eigenvalues of a graph

if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that ...

**2**

votes

**1**answer

38 views

### Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?

Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ...

**5**

votes

**1**answer

151 views

### Dimensions of a vector space akin to modular symbols

The group $\operatorname{SL}_2(\mathbb Z)$ acts on polynomials in two variables $\mathbb C[x,y]$ via $A\cdot f(x,y)\mapsto f(A^{-1}.(x,y))$ where $(x,y)$ is regarded as a column vector. There are two ...

**0**

votes

**0**answers

31 views

### Changes in singular Values of matrix when adding row

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is:
Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...

**0**

votes

**0**answers

21 views

### Determinant of special diagonal matrix [migrated]

Suppose A is a square matrix of any order whose diagonal entries are integers which are pairwise prime and all other entries are 1. Is there any method to find determinant of this matrix A.

**3**

votes

**0**answers

71 views

### Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...

**2**

votes

**2**answers

123 views

### Matrix inequality

Let $\mathbf{Z,R}$ two Hermitian semidefinite positive matrices with all eigenvalues larger than one. Intuition drives me that
$\mathbf{R}^{-1/2}\mathbf{Z} \left(\mathbf{R}^{-1/2}\right)^H - ...

**5**

votes

**0**answers

83 views

### A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...

**0**

votes

**0**answers

14 views

### On the stability analysis of a discrete difference system with multiplicative noise

If we assume that
\begin{equation*}
\rho \{\phi \otimes \phi+\psi \otimes \psi\}<1
\end{equation*}
where $\rho$ denotes the spectral radius, then can we verify the following inequality holds
...

**1**

vote

**1**answer

44 views

### Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105
Unfortunately I am struggling to make the algorithm work on ...

**-1**

votes

**1**answer

188 views

### Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.

**2**

votes

**2**answers

129 views

### What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null

Let $\mathbb F_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces ...

**1**

vote

**0**answers

142 views

### Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...

**1**

vote

**0**answers

51 views

### What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...

**5**

votes

**1**answer

151 views

### The Maximal $\ell_2$ norm of a signed sum of vectors

Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors:
$$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$
where $s_j$'s can only take values of $+1$ or $-1.$ I ...

**0**

votes

**1**answer

65 views

### Scalar restriction and scalar extension

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h_*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^*\colon{\sf ...

**0**

votes

**0**answers

54 views

### About bounding values of quadratic forms

It would be helpful if someone can share (either as references) examples of calculations/analysis which achieves bounding of values of quadratic forms in say either of the following situations,
...

**1**

vote

**1**answer

62 views

### Traces and projectors

Suppose that $V$ is a $\mathbb C$-vector space. I'm eventually interested in the infinite-dimensional case, but let's say for now that it's finite dimensional. Suppose that $\mathscr S$ is a ...

**4**

votes

**2**answers

193 views

### Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$

Let $M$ be an invertible symmetric $2n \times 2n$ matrix with entries in the finite field $\mathbb{F}_2$. Is $\mathrm{Ker}\ (M^2 - I_{2n})$ necessarily even dimensional?

**1**

vote

**1**answer

41 views

### proving that a smooth curve in Euclidean n-space contains n+1 affinely independent points

If I let $f(\theta)=((\mathrm{cos} \theta)X+(\mathrm{sin} \theta)Y)^{n-1}$ and view the range of this curve as a subset of the space of homogeneous polynomials of degree $n-1$ in two variables viewed ...

**4**

votes

**2**answers

97 views

### Convexity of a (non-symmetric) function of matrices

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the ...

**0**

votes

**0**answers

53 views

### Matrices over a finite field with given Jordan normal form over the algebraic closure [migrated]

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...

**0**

votes

**0**answers

21 views

### Alternative form for weighted least squares [migrated]

Coefficients $\beta$ can be estimated from $y$ by weighted least squares with:
$ \beta = (X^T\Sigma^{-1}X)^{-1} X^T \Sigma^{-1} y $
where $\Sigma$ is the covariance matrix of the noise.
Let $N$ be ...

**2**

votes

**0**answers

249 views

### Good covering of a sphere

Consider a sphere $S_r(0)$ with center at zero and radius $r$ in the Hamming space $\{0,1\}^n$.
We will be interested in covering this sphere with balls of radius $\rho < r$.
We know that there ...

**2**

votes

**1**answer

65 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...

**1**

vote

**2**answers

134 views

### Inverse of a matrix expression

Let
$$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$
where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements.
Is there a way to simplify this expression in order to ...

**-1**

votes

**0**answers

31 views

### A smooth family of symplectic forms [migrated]

Let $A(t)\in\mathbb R^{2n\times 2n}$ be a smooth family of nondegenerate skew-symmetric matrices, $t\in\mathbb R$. Then $A(t)$ represents the family of symplectic forms $\omega_t(u,v)=\langle ...

**1**

vote

**0**answers

28 views

### Can we give efficiently the solution of a bilinear system of equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations
$$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$
where $\alpha=(\alpha_1,...,\alpha_s)$ and ...

**3**

votes

**1**answer

83 views

### Probe permutationally matrix extreme properties

Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$.
Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal ...

**3**

votes

**1**answer

449 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

**1**

vote

**0**answers

44 views

### How to define the determinant of a morphism between graded Lie algebras?

I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...

**-1**

votes

**0**answers

66 views

### Minimum rank non-negative matrix summations

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$ of rank $r$.
What is minimum $k$ such that
$$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains $R,S$ ...

**1**

vote

**0**answers

40 views

### Probability of non-negative matrix relaxation

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, take $\mathscr{M}[M]=\{Q\in R_{\geq0}^{n\times n}:Q[ij]>0\iff M[ij]=1\}$.
Does ...

**3**

votes

**1**answer

81 views

### Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...

**9**

votes

**0**answers

292 views

### A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...

**3**

votes

**1**answer

92 views

### Upper bounds on elements of a matrix

During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...