**11**

votes

**0**answers

240 views

### Who stated and proved the “Hopf lemma” on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.
Nondegenerate here means ...

**2**

votes

**1**answer

138 views

### Can sparse matrices satisfy the Null Space Property?

Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if
$$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ...

**3**

votes

**1**answer

200 views

### Reconstructing a (unitary) matrix from the determinant of its sub-matrices

I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ ...

**0**

votes

**1**answer

137 views

### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...

**1**

vote

**0**answers

161 views

### Incoherence of the row/column span

Due to V.Chandrasekaran., et al (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that:
$$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$
where the lower bound is achieved (for ...

**2**

votes

**0**answers

45 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 ...

**7**

votes

**2**answers

848 views

### Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...

**3**

votes

**0**answers

123 views

### Kasteleyn, Gessel-Viennot and eigenvalues

The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...

**6**

votes

**0**answers

301 views

### On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots,
f_k(x) \in \mathbb F_q[x]$ be ...

**4**

votes

**1**answer

230 views

### Is there a generalization of linear algebra that allows fractional ranks?

The rank is the number of linearly independent rows/cols of a matrix. Generally, we think of linear independence as a binary property. But we could imagine an alternative definition that allows for ...

**1**

vote

**2**answers

200 views

### Möbius transformation by 3 points in the Minkowski model

Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...

**11**

votes

**4**answers

529 views

### Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 ...

**-1**

votes

**1**answer

115 views

### Inequality between two matrices

Given a full rank $n \times m$ matrix $K$ with $m<n$ and an invertible symmetric matrix $J$. Let $A$ be a symmetric positive semi-definite $n \times n$ matrix such that
\begin{equation}
(K^T ...

**0**

votes

**1**answer

59 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

130 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

193 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

**3**answers

833 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

252 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

189 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

130 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

65 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

198 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

120 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

58 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

118 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

321 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

141 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

104 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

205 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

92 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

258 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

497 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

149 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

173 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

192 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

160 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

229 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

64 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

287 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

95 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

107 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

88 views

### How could I prove this equality 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

236 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

44 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

319 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

252 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

160 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 ...

**0**

votes

**1**answer

138 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 ...

**3**

votes

**1**answer

122 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 ...