**4**

votes

**2**answers

163 views

### Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...

**6**

votes

**0**answers

180 views

### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$
It is NP-hard to compute $S_M$ exactly I believe by applying the ...

**1**

vote

**1**answer

149 views

### Is there a nice choice-free argument to count the number of sublattices?

It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$.
Here is a proof of this fact:
Proof: choosing a basis of ...

**10**

votes

**2**answers

418 views

### How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i=
\biggl(\begin{matrix}
C_i+E_i & B_i \\
B_i^T & D_i-F_i
\end{matrix} \biggr)
$, where $B_i$ is an arbitrary $n\times n$ ...

**1**

vote

**0**answers

51 views

### Directed graph Laplacian with exactly one negative eigenvalue

Let $G$ be a digraph with adjacency matrix $A =(A_{ij})$ where $A_{ij}=1$ if and only if there is a directed edge $i \to j$ and $A_{ij}=0$ otherwise. Let $D= (D_{ij})$ be the degree matrix with ...

**-1**

votes

**1**answer

117 views

### How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? [closed]

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...

**2**

votes

**1**answer

163 views

### Extracting a full rank matrix from a 0-1 matrix

If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of ...

**10**

votes

**0**answers

366 views

### Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...

**2**

votes

**1**answer

76 views

### Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually, I'm not exactly looking into bipartite but my ...

**3**

votes

**1**answer

153 views

### Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$
Let $M_n$ be the space of $n \times n$ real matrices.
Question:
For which $n$, is there an inner product on $M_n$ which satisfies:
$$(*) \, \, \langle Q^TXQ,Q^TYQ ...

**4**

votes

**2**answers

243 views

### Derivative of eigenvectors of a matrix with respect to its components

Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 ...

**6**

votes

**1**answer

191 views

### Fast Symbolic Linear Algebra CAS?

I am a regular user of Mathematica, Julia, and MATLAB but I am looking for something different. The problem I am trying to solve in Mathematica only requires (dense) linear algebra to specify but is ...

**0**

votes

**1**answer

129 views

### Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?

Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$.
In "Iskovskikh ...

**2**

votes

**2**answers

157 views

### Intersection of Subspaces with $O(3)$

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below.
For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...

**2**

votes

**1**answer

153 views

### programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism

Let the integers $n\geq 2$, $k\geq 1$, $v=0$ or $1$ and $n_1,\cdots,n_k\geq 1$ such that
$$
\sum_{i=1}^k n_i+v=n.
$$
Define $P_a^b=0$ if both $a,b$ are odd and $P_a^b={{[a/2]}\choose {[(a+b)/2]}}$ ...

**1**

vote

**0**answers

44 views

### Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with ...

**5**

votes

**2**answers

209 views

### Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups

I asked this question before at Math.SE (link) but got no answer.
Let $G$ be an algebraic group over an algebraically closed field $k$ of characteristic $p \geq 0$. Then any parabolic subgroup $P$ ...

**4**

votes

**2**answers

130 views

### Collection of projection operators in finite dimension and algebraic techinques

Consider a set of linearly independent vectors $\{x_1,\dots,x_n\}$ in some finite-dimensional Hilbert space $H$. For any subset $S \subset [n]$, let $P_S$ be the (orthogonal) projection (operator) ...

**0**

votes

**0**answers

75 views

### the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group

Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...

**0**

votes

**1**answer

89 views

### Example distance metric that is not conditionally negative definite

Theorem 4.1 of this paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an ...

**0**

votes

**1**answer

244 views

### Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?

Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets
...

**4**

votes

**1**answer

82 views

### Kolmogorov complexity for matrices

In applications one often encounters very large matrices that barely fit in computer memory, if at all. Naturally one wishes to represent those matrices as compactly as possible. Sometimes one even ...

**9**

votes

**1**answer

185 views

### Exact determinant of a circulant matrix

The wikipedia gives us a formula for the determinant of a circulant matrix. That is:
$$\mathrm{det}(C)
= \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \dots + c_1\omega_j^{n-1})= ...

**1**

vote

**1**answer

115 views

### Determinant of discrete Laplacian

It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix
$$\begin{pmatrix}
2 & -1 & & \\
-1 & 2 & \ddots & \\
& ...

**4**

votes

**2**answers

206 views

### Why are the vectors with this special structure linearly independent with high probability?

Given $a_i\in\mathbb{R}^m$ for $i=1,\ldots,2m$ with independent and identically random entries of some continuous distribution. Every choice of $m$ vectors from $\{a_1\ldots,a_{2m}\}$ is linearly ...

**11**

votes

**1**answer

444 views

### An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...

**5**

votes

**2**answers

214 views

### Induced matrix norm less than one for matrices with spectral radius less than one

Let $A$ be a square matrix with elements in $\mathbb{R}$ or $\mathbb{C}$,
$\rho\left(A\right)$ stands for the spectral radius of $A$, i.e.,
the maximum absolute eigenvalue of $A$; $A^{*}$ is the ...

**0**

votes

**0**answers

34 views

### If $A$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$

If $A \in M_n$ is doubly stochastic and reducible.
Why is $A$ permutation similar
to a matrix of the form $A_1 ⊕ A_2$, in which both $A_1$ and $A_2$ are doubly stochastic?

**1**

vote

**0**answers

75 views

### Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?

Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix
$M = \frac{1}{2}(A + {A^T})$.
Why does $\rho (A) \le {\lambda _{\max }}(M)$?

**1**

vote

**1**answer

79 views

### Multiplicatively closed subsets of $\mathbb{C}^{n \times n}$

While dealing with another problem I saw that I need to classify subspaces $V$ of $\mathbb{C}^{n \times n}$ that are multiplicatively closed.
So to get an idea of the nature of the subspaces I ...

**0**

votes

**1**answer

116 views

### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...

**1**

vote

**1**answer

76 views

### $0/1$ programming multiple quadratic constraints

If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time.
If we have an $n$-variable degree $2$ system how many constraints ...

**0**

votes

**2**answers

432 views

### Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...

**1**

vote

**0**answers

17 views

### How to find a subset of a matrix that has minimum condition number? [duplicate]

Suppose matrix $A$ is consist of M column vectors, how can we find a subset $B$, consisting of N column of $A$ (N＜M), that has minimum condition number (the ratio of maximum singular value by minimum ...

**0**

votes

**0**answers

55 views

### An extremal combinatorics problem

What is the minimum rank $r$ of an $n\times n$ square positive integer matrix such that sum of entries of every $\sqrt r\times\sqrt r$ submatrix is distinct and such that difference between minimum ...

**1**

vote

**0**answers

53 views

### How to associate the following two kinds of real polynomials?

Suppose the following real polynomial of $n$ variables
$$f(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}a_IX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$$
is easy or familiar to us, but I need to deal with ...

**1**

vote

**0**answers

73 views

### A question on Perron–Frobenius theorem [closed]

Let $A \in M_n$ is nonnegative(all $a_{ij}\ge0$).
Suppose $A$ has a nonnegative eigenvector(all entries$\ge0$ ) with $r ≥ 1$ positive entries and $n − r$ zero entries.
Why is there a permutation ...

**4**

votes

**1**answer

133 views

### subset of hermitian matrices given by eigenvalues form a submanifold

Let $\mathcal{O}_\lambda$ be the set of hermitian $n+1 \times n+1$ matrices with Eigenvalues $\lambda = (\lambda_1, \dots, \lambda_{n+1})$.
and $\mathcal{O}^\mu$ the set of hermitian $n \times n$ ...

**6**

votes

**1**answer

194 views

### Bounds for maximum determinant of circulant matrices

The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns.
An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...

**2**

votes

**1**answer

97 views

### Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic,
linear operators, which says the following:
Defintion:
Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...

**6**

votes

**1**answer

457 views

### Why is $(A^\perp)^\perp = A$?

On page 52 of this paper, Iwasawa considered the bilinear symmetric non-degenerate pairing $\Phi_n \times \Phi_n \rightarrow \mathbb{Q}_p/\mathbb{Z}_p$ defined by
$$\langle \alpha, \beta \rangle_n := ...

**1**

vote

**1**answer

214 views

### Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...

**11**

votes

**4**answers

868 views

### Probability two products are equal

I am interested in the following simple looking problem on which I am stuck. Let $M$ be a fixed $m$ by $n$ matrix with $\pm1$ elements. Let $x$ and $y$ be two independently sampled random ...

**10**

votes

**2**answers

279 views

### Why Householder reflection is better than Givens rotation in dense linear algebra?

It’s obvious that Givens rotation works better with sparse matrices. But I don’t know why Householder reflection is better for dense matrices. Does it require less computations? Or it’s numerically ...

**3**

votes

**1**answer

204 views

### Why is dual lattice a lattice, in the context of complex tori

I have a simple linear algebra question regarding the definition of dual of a lattice; it was asked by someone else here three months ago on mathstackexchange but got no answer and few views, so ...

**0**

votes

**0**answers

248 views

### When does this matrix have full rank?

Suppose $\mathbf{B}\in\left[0,1\right]^{T\times M}$ is a binary matrix,
$\mathbf{B}_{i}$ is a column of $\mathbf{B}$, and $\mathbf{X}\in\mathbb{R}^{N\times T}$
is a matrix where the columns are ...

**2**

votes

**0**answers

122 views

### Which functions preserve the connectivity of graphs/components?

I am somewhat stuck working on an issue and would really love some guidance. I will state the problem, my current state and what led to it in case the solution lies beyond where I was looking
The ...

**0**

votes

**2**answers

83 views

### Simultaneous special orthogonal similarity problem

Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
...

**0**

votes

**0**answers

66 views

### Simplifying product of matrix exponential?

Is there a known generalization for n-term matrix exponential multiplication?
I am aware that the Baker–Campbell–Hausdorff formula could be used, e.g.:
...

**0**

votes

**0**answers

98 views

### A linear combination problem

Given $0/1$ $n\times n$ matrix $M$.
Suppose we have $2$ vectors $\lambda,\mu\in\Bbb R^{1\times n}$ such that both
$$\lambda M\in\{0,1\}^{1\times n}$$
$$M\mu'\in\{0,1\}^{n\times 1}$$
holds with $'$ ...