**0**

votes

**1**answer

53 views

### Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...

**1**

vote

**1**answer

108 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**4**

votes

**1**answer

130 views

### Eigenvalues of generalized Vandermonde matrices

Given a strictly increasing sequence $0<x_1<x_2<\dots<x_n$ of $n$ strictly positive real numbers and a second strictly increasing sequence $e_1<\dots e_n$
of $n$ real numbers, the ...

**17**

votes

**2**answers

698 views

### Example of a space for which $V \cong Hom(V,V)$

Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology.
Is there a non-trivial ...

**9**

votes

**1**answer

201 views

### Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle

Let $z_{1},\dots,z_{k}$
be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$
consider the rectangular Vandermonde matrix
$$
V_{N}=\begin{pmatrix}1 ...

**2**

votes

**1**answer

165 views

### Angle between two subspaces

Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space
$T_xM=E^s(x)\oplus ...

**0**

votes

**1**answer

111 views

### Constructing a continuous matrix valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...

**2**

votes

**1**answer

61 views

### Relating joint probability to norm of vector of probabilities

I have a set of Bernoulli random variables $X_1,X_2,\ldots, X_n$ and I would want to bound the joint probability $P(X_1=0,X_2=0,\ldots, X_n=0)$ using the norm $\lvert \lvert \mathbf{p}\rvert \rvert$, ...

**2**

votes

**1**answer

188 views

### Powers of linear functions span the space of polynomial functions?

Let $P_n$ be the space of polynomials in $n$ variables over a field of characteristic 0.
I'm very sure that the space spanned by powers of linear functions is the whole space $P_n$.
Anyone can come ...

**0**

votes

**0**answers

108 views

### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...

**0**

votes

**0**answers

55 views

### Union of subspaces not contained in algebraic variety

Consider an $n \times n$-matrix $A$ and an $m \times n$-matrix $C$ where $m < n$.
I am asking myself whether it is possible to formulate an algebraic condition on $A$ and $C$ such that the union
...

**2**

votes

**1**answer

104 views

### Union of orthogonal complements of subspaces is not contained in a proper algebraic variety

Consider an $n \times n$-matrix $A$ and an $m \times n$-matrix $C$ where $m < n$. For each $t \ge 0$ the kernel $\ker Ce^{At}$ is a (say $k$-dimensional) subspace. Suppose the intersection of these ...

**6**

votes

**3**answers

310 views

### On the rank of a matrix $S$ with coefficients in $\mathbb F_{2^m}$

Let $\mathbb F$ be a finite field with characteristic 2 and let $S \in M(2k, 2k, \mathbb F)$ be the matrix defined as follows
$$ S=\left[\begin{array}{ccccccc}
0 & ...

**3**

votes

**0**answers

109 views

### Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle ...

**0**

votes

**1**answer

97 views

### Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as
\begin{align}
...

**0**

votes

**0**answers

142 views

### Proving that a certain matrix inverse is always positive definite

Take any positive definite Hermitian matrix $X$.
Put all values of $X$ equal to 0 except for the values along the center $2K+1$ diagonals which are kept untouched. Denote the new matrix by $Y$.
Let ...

**0**

votes

**2**answers

88 views

### Union of linear inequalities cover whole space?

We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$.
There are $k$ sets of linear inequality constraints on the $a_i$.
I need to check that any choice of $a_i$ satisfies at least ...

**1**

vote

**0**answers

251 views

### how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
...

**10**

votes

**2**answers

483 views

### A binomial determinant fomula

Is there an existing or elementary proof of the determinant identity
$
\det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1
$?

**2**

votes

**3**answers

114 views

### Applications of rank factorization or full rank decomposition [closed]

I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r ...

**3**

votes

**1**answer

97 views

### For a linear dynamic system, what can we learn from its singluar value and rank?

Given a linear system $\frac{dx}{dt}=Mx$, what's the relationship between the dynamic's property and the singular value decomposition/rank of $M$ ?

**2**

votes

**2**answers

164 views

### Obstructions to Smith normal form of a special type

Suppose I have a Smith normal form $S,$ and I want to have an $M \in SL(n, \mathbb{Z}),$ such that $M - I$ has SNF $S.$ Is this always possible? For a (potentially) somewhat harder question, what if I ...

**3**

votes

**1**answer

169 views

### Equivalence of Hadamard Graph and Hadamard Matrix

I'm reading Distance Regular Graphs by Brouwer, Cohen, and Neumaier. In section 1.8, they explained Hadamard graphs.
Conversion from a Hadamard Matrix into a Hadamard Graph
An $n$-Hadamard graph $G$ ...

**1**

vote

**1**answer

157 views

### A linear algebraic q-difference equation [SOLVED]

I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...

**2**

votes

**3**answers

202 views

### LU decomposition

Consider a $N \times N$ symmetric real matrix $A$: $A_{ij} = (\sum_{k=1}^N n_{ik}) \delta_{ij} - n_{ij}$, where $n_{ij}$ is a real symmetric matrix whose elements are equal to $1$ or $0$. $A$ has one ...

**2**

votes

**0**answers

59 views

### System of 2 linear q-difference equations with singular matrix

I would like to solve the following algebraic linear system of q-difference functional equations:
\begin{cases}
...

**4**

votes

**0**answers

285 views

### An oddity i found in some linear equations. (I'm in 7th grade.) [closed]

Okay, so I've started Algebra I this year, and i've always had a love for math. And at one point in the course we were presented with an equation similar to this one:
5x + 3 = 8x + 3
And so I solved ...

**4**

votes

**1**answer

88 views

### Weak ergodicity of nonhomogenous products of 0-1 matrices

Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...

**0**

votes

**1**answer

84 views

### Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...

**0**

votes

**0**answers

67 views

### How could I prove this eduality for eigenvalues of Laplacian matrix?

I would be glad if you have some comments that how I could prove following statement.
Suppose that graph $G =(N, E)$ be given. The the following program computes the $k$-smallest eigenvalues of the ...

**6**

votes

**1**answer

216 views

### Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...

**2**

votes

**0**answers

42 views

### Nullspace of a matrix modulo an ideal

Suppose $R$ is a multivariate polynomial ring and $I$ is an ideal in $R$.
Let $M$ be a $n\times n$ square matrix with entries in $R$, and suppose that det($M$) lies in $I$.
Thus, $M$ has a ...

**10**

votes

**2**answers

285 views

### Determinant and eigenvalues of a specific matrix

This came up in a conversation with an engineer friend of mine.
Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries
$$
A_{ij} = e^{-c(i-j)^2}.
$$
Is there a name for this ...

**3**

votes

**2**answers

245 views

### norm of the matrix series

The goal is to obtain an upper bound for the norm of the vector
$$
\left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\|
$$
for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ...

**1**

vote

**1**answer

144 views

### Probabilistic Johnson-Lindenstrauss Lemma for arbitrary points

Consider the following standard formulation of the Johnson-Lindenstrauss lemma:
Lemma (JL).
For any $0<\epsilon < 1$ and any integer $n$, let $k$ be a positive integer such that $k\geq ...

**-1**

votes

**1**answer

126 views

### Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows:
For each weight $\mu$, let $L(\mu)$ be the irreducible ...

**2**

votes

**1**answer

105 views

### Non-orthogonal vectors cosine-enhancing transformation

Let consider a real vector space $\mathbb{R}^n$ of dimension $n$, where $\langle \cdot, \cdot\rangle$ and $||\cdot ||$ are the standard inner product and $\ell_2$ indeced norm.
Does there exist a ...

**-2**

votes

**1**answer

589 views

### Is there such a thing [closed]

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le5$, are there
$W\in\mathcal{U}\left(4\right)$ and nontrivial $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$, such that ...

**5**

votes

**1**answer

279 views

### History of Jordan Canonical Form?

Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in:
When and how was it first stated? (I understand it was independently stated ...

**2**

votes

**1**answer

320 views

### Is this projective variety empty?

Let $\mathbf{a}\in\mathbb{R}^{4}$ and the entries of a $3$ by $4$ real matrix $A$
be the variables of the polynomial equations, $\det\left(A\right)=0$, $S_{1},S_{2}\subseteq\left\{ 1,2,3,4\right\}$
...

**0**

votes

**1**answer

169 views

### Nonnegative Matrix

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ...

**3**

votes

**3**answers

102 views

### For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?

In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true:
Let $G$ be a group, $g\in G$, and $\rho:G \to ...

**7**

votes

**1**answer

85 views

### Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity

I'm currently trying to get familiar with the Jordan normal form for matrices; and after some example I ask for the possible Jordan-form for the Carleman matrix for the function $f(x) = \sin(x)$ when ...

**0**

votes

**1**answer

75 views

### Measuring the extent of entanglement in tensor products

Is there a non-negative integer valued grading function on the tensor product of two Hilbert spaces which measures the extent of entanglement ?

**1**

vote

**1**answer

105 views

### Form of a block upper triangular matrix of finite order

If I take a block upper triangular matrix and the matrix is known to be diagonalizable and the leading blocks in the diagonal are each of finite order, is it true that away from the leading block ...

**3**

votes

**1**answer

74 views

### Set of orthogonal simplexes or partial mutually unbiased bases

I am interested in the existence of a set of vectors $\{ v_{ij} \}_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^*_{ij} v_{ij'} \right\vert = ...

**2**

votes

**1**answer

101 views

### Ranks of higher incidence matrices of designs

In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is
$$
Rk_{2}(N)=v-(d_{p}+1),
$$
where $d_{p}$ is the ...

**0**

votes

**2**answers

154 views

### Does there exist a linear equation system with a NAND-like behavior? [closed]

Does there exist an equation system with only linear equalities and inequalities and exhibits a NAND-like behavior?
By 'NAND-like' I mean, among all the variables in the system there are 3 vars $x_1$, ...

**1**

vote

**0**answers

98 views

### Solve a linear equation with many variables using only 1 and -1

For a program I am writing, I would find it useful to find the least possible positive solution to a linear equation, using only -1 and 1 for roots.
For example...
...

**0**

votes

**0**answers

63 views

### Interpret some coefficients in algebras

Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...