**4**

votes

**0**answers

291 views

### A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
...

**1**

vote

**1**answer

106 views

### Compressing a system of linear equations

Consider the system of linear equations $A\mathbf x=\mathbf b$ in which $A$ is an $m\times n$ matrix with $m < n$ and with the following property:
Property $\Gamma$: Given $M=\{ M_1,\cdots,M_r ...

**2**

votes

**1**answer

197 views

### A similar Cauchy-Schwarz inequality with linear-algebra

Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction.
Assume that $A$ and $B$ are contractions such that
$I-AA^*$ and $I-BB^*$ are ...

**0**

votes

**1**answer

60 views

### Reference request for: inverse of a non-singular M-matrix has all elements non-negative?

Does anyone know the best (earliest?) reference please for the proof that the inverse of a non-singular M-matrix has all elements non-negative?

**0**

votes

**0**answers

33 views

### Nonlinear matrix equation (transpose) [duplicate]

Let $H$, $M$ and $N$ be 10 by 10 matrices over the integers. If $M$ and $N$ are known, how do you solve for $H$ from the following equation?
$M = H N H^T$
where $H^T$ is the transpose of H.

**1**

vote

**0**answers

118 views

### Reduce a Combinatorial problem

It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...

**2**

votes

**2**answers

232 views

### Relationship between largest eigenvalue of a positive matrix $A$ and $A∘A^T$

I'm wondering whether there is certain relationship between the largest eigenvalue of a positive matrix(every element is positive, not neccesarily positive definite) $A$, $\rho(A)$ and that of ...

**0**

votes

**1**answer

143 views

### Kneser graphs eigenvalues

Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...

**5**

votes

**1**answer

169 views

### Comparing Krein-Rutman theorem and Perron–Frobenius theorem

Krein–Rutman theorem is a generalization of Perron–Frobenius theorem, I know that things could be more subtle in infinite dimension, yet there's an important result in Perron–Frobenius that's missing ...

**2**

votes

**1**answer

79 views

### Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...

**1**

vote

**1**answer

59 views

### Extending a matrix with a certain property over a finite field

Suppose we have a $m \times n$ matrix $M$ with $n > m$ with entries over a finite field, say $\mathbb{F}_q$ with $q$ considered to be large compared to $m,n$. Suppose that $M$ has the property that ...

**3**

votes

**0**answers

121 views

### Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!

**3**

votes

**3**answers

538 views

### Algebraic K-theory can be seen as a generalization of Linear algebra? [closed]

Algebraic K-theory can be seen as a generalization of Linear algebra?
If yes, how so?

**8**

votes

**1**answer

120 views

### A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...

**2**

votes

**0**answers

131 views

### Cavalieri's principle and inversion of the Vandermonde matrix

There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...

**0**

votes

**0**answers

117 views

### classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...

**7**

votes

**2**answers

399 views

### Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it ...

**1**

vote

**0**answers

138 views

### Is finding a single vector in the null space as difficult as discovering the whole null space?

Let $A \in \mathbb R^{k\times n}$ be a matrix of rank $k$, where $k \ll n$. One can use Gaussian eliminations to discover $\operatorname{null}(A)$ at the cost of $O(nk^2)$. My question is:
Is the ...

**3**

votes

**0**answers

92 views

### Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem.
Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that
$$ \| D_1 A ...

**0**

votes

**0**answers

64 views

### Heat equation with graph laplacian

I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix.
Suppose we have a connected graph with unknown temperature on vertices. ...

**1**

vote

**1**answer

122 views

### Position of complete flags

$E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F_1,F_2,\dots,F_n)$ and $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are two complete flags of $E$.
We say that $(\mathbb{F},\mathbb{G})$ is in ...

**0**

votes

**0**answers

17 views

### C-Periodic boundary conditions

I'm working with linear chain of strongly correlated electrons.
These types of models have problems due to finite size effects, this leads
one to consider C-Periodic boundary conditions as an attempt ...

**4**

votes

**1**answer

278 views

### dim Hom(V,W) =?

I asked this question on Mathematics Stack Exchange, but got no answer:
Given two vector spaces $V$ and $W$ over a field $K$, what is the dimension of $\operatorname{Hom}_K(V,W)\ $?
To state the ...

**13**

votes

**1**answer

381 views

### A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space

While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
Question: Do there exist a non-trivial ...

**1**

vote

**0**answers

33 views

### Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...

**7**

votes

**1**answer

273 views

### What is the total polarization of the determinant?

Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the total polarization ...

**2**

votes

**3**answers

233 views

### Solving a quadratic matrix equation with non-squared matrix

I was trying to solve the problem of finding the value of a non-squared matrix $T$ ($n \times m$) which solves
$$ T^T T = X$$
where $X$ is a symmetric and positive semidefinite $m \times m$ matrix, ...

**0**

votes

**1**answer

161 views

### Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...

**0**

votes

**0**answers

86 views

### Perturbed linear system, particular form

We have a linear system $Ax=b$ where $A$ is real and symmetric, all elements of its main diagonal are strictly positive and all off-diagonal elements are $\leq 0$. Further, $A_{ii} > -A_{ij} \; ...

**6**

votes

**2**answers

305 views

### $\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le ...

**1**

vote

**0**answers

68 views

### spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e.,
...

**1**

vote

**0**answers

71 views

### Bounds for the infinity norm of the inverse for certain diagonaly dominant matrices

I m trying to analyse the stability against perturbations for a specific system of linear equations $Ax=b$.
For this, i use the standard condition number $||A||_{\infty}||A^{-1}||_{\infty}$.
Here ...

**3**

votes

**1**answer

131 views

### eigenvalues of product of many symmetric positive definite matrices

Given $A_1, ..., A_n$ ($n\geq 3$), where each $A_i$ is a $d$-by-$d$ symmetric, positive definite matrix, define $S = A_1\cdot A_2\cdot...\cdot A_n$ (product of all the $A_i$'s). Let $\lambda_1(A)$ and ...

**3**

votes

**1**answer

357 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

**1**

vote

**1**answer

44 views

### Maximising a Rayleigh quotient over a subspace II

Let $M$ be an SPD matrix and let $\Pi=QQ^T$ be the orthogonal projection onto the range of $Q$ (a "tall" matrix with orthonormal columns). I have an expression in the form
$$\tag{1}
...

**2**

votes

**0**answers

95 views

### Hilbert Schmidt Operators and the Conditional Expectation Operator

Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...

**0**

votes

**0**answers

52 views

### Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$.
We fix the value of one variable $q_i \in (0,1)$, then maximize ...

**2**

votes

**0**answers

59 views

### Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if ...

**1**

vote

**1**answer

65 views

### Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...

**2**

votes

**0**answers

33 views

### Metabolic vs stably metabolic

Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ ...

**3**

votes

**0**answers

102 views

### Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows:
For a ...

**0**

votes

**1**answer

80 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**1**

vote

**0**answers

26 views

### Heuristic for choosing n-vectors from n-sets

my given problem is:
choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...

**2**

votes

**1**answer

72 views

### Triangular Smoothing Formula Optimization

I'm using a 5-point Triangle Moving Average:
$$S_j = (Y_{j-2} + 2Y_{j-1} + 3Y_j + 2Y_{j+1} + Y_{j+2}) / 9$$
The problem is that I often need to smooth my data more than once, and when I do this too ...

**0**

votes

**0**answers

40 views

### Spanning Hadamard product powers (Schur products) in Euclidean space

This is a question I asked last year over at stackexchange. Got no answer, it might be sufficiently non-trivial to justify bringing it into here.
Fix two $k$-vectors $\mathbf u$ and $\mathbf v$, and ...

**3**

votes

**4**answers

337 views

### Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...

**4**

votes

**0**answers

88 views

### A question on hyperplanes in partial linear spaces and hypergraphs

A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear ...

**0**

votes

**1**answer

64 views

### Decompositions of sparse symmetric matrices and methods for solving large linear equations

I am writing code for solving linear equations of the form
$$A_{n\times n}\cdot x=1_n$$
where $n$ is on the order of $10^6$ and $A$ is a symmetric matrix with approx $10^3$ nonzero entries in each ...

**6**

votes

**2**answers

239 views

### Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.
Does an upper ...

**5**

votes

**1**answer

174 views

### Tomographic reconstruction of a point set from projections

Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align*}
S = \{x_1, \dots, x_N\} \subset \mathbb R^n.
\end{align*}
Let $v \in \mathbb R^n$ and consider the image set (not counting ...