Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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**2**answers

44 views

### Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets

$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...

**1**

vote

**0**answers

99 views

### Matrix rescaling increases lowest eigenvalue?

Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...

**9**

votes

**1**answer

501 views

### Relation between the Axiom of Choice and a the existence of a hyperplane not containing a vector

In a lot of problems in linear one uses the existence, for each $E$ vectorial space over a field $k$ , and each $x\in E$, of a Hyperplane $H$ such that $E=k\cdot x \oplus H$ (Let us denote $\mathcal{P}...

**0**

votes

**0**answers

17 views

### Can we build the simplex Tableau from a graphical point (vertex)?

Everything is in the title, starting from a vertex in 2D dimension, can we continue the simplex method from there, using the vertex coordinates in the simplex tableau ?
Thanks a lot for your time

**6**

votes

**1**answer

156 views

### Is this bound uniform in $N$?

I encountered this small combinatorial problem and do not quite know how to solve it:
Consider a set $\mathbf N:=\left\{1,2,....,N \right\}.$ This set has $\binom{N}{2}$ many subsets of cardinality $...

**2**

votes

**0**answers

51 views

### Order relation between cohomology groups

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex
$$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...

**2**

votes

**1**answer

59 views

### On submatrices: size bound

Let $M$ be a generic $2n\times 2n$ matrix and fix $k\leq n$.
Suppose $\mathcal{F}$ is a family of submatrices under the conditions that $A\in\mathcal{F}$ provided
(a) $A$ is a $k\times k$ ...

**3**

votes

**0**answers

66 views

### Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...

**1**

vote

**1**answer

64 views

### Closed Form Solution for Optimization Problem over the Space of Rigid Transforms

Is there a closed form solution to this constrained optimization problem:
\begin{equation}
\min_{R \in SO(3),\, \mathbf t \in \mathbb R^3} = \sum_{i = 1}^N \| M_i(R \mathbf p_i + \mathbf t) \|^2,
\...

**3**

votes

**1**answer

161 views

### Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?

Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...

**4**

votes

**1**answer

146 views

### Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...

**-1**

votes

**0**answers

52 views

### Obtaining a Relation Between Positive Matrices

Consider $n \times n$ non-negative binary matrices
${\bf A}_i$ with $1\leq i \leq m$ over $\mathbb{R}$.
Assume that $1\leq k \leq n$ is selected as a fixed number.
Suppose that a subset of size $k$...

**8**

votes

**3**answers

242 views

### Matrix determinant inequality proof without using information theory

Let $A$ be a $k \times n$ orthogonal matrix; i.e., $AA^T = I_{k \times k}$. For $1 \leq j \leq n$, let the squared norm of the $j$-th column of $A$ be denoted by $\alpha_j^2$; i.e.,
$$\sum_{i=1}^k a_{...

**-2**

votes

**1**answer

125 views

### sum of positive definite matrix

sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$
can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i....

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votes

**0**answers

67 views

### How to understand the span of a matrix? [closed]

https://people.eecs.berkeley.edu/~brecht/cs294docs/week7/12.candes.recht.pdf
In the section 3.3 of the paper above, the author explains the span of matrices:
To apply our results to recovering low-...

**6**

votes

**0**answers

75 views

### Geometric mean of three or more positive definite matrices

The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...

**2**

votes

**0**answers

33 views

### Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?

Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...

**1**

vote

**1**answer

180 views

### fiber of a map into Grassmanian

Suppose $R\subset K=K_0\supset K_1\supset K_2\supset...\supset K_{n-1}\supset K_n=\{0\}$ are all vector spaces with $\dim R\cap K_i=r_i$ where $r_i$ are some fixed numbers. Suppose $O\subset Gr(r_0,\...

**2**

votes

**2**answers

234 views

### Laplacian of an infinite graph and connected components

For a finite graph with undirected, unweighted edges, a well-known result is that the dimension of the null space of the Laplacian matrix gives the number of connected components. Does this result ...

**15**

votes

**3**answers

541 views

### $\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$

I'm interested in
$$\sup_{x, y} \left\| A x + B y\right\|_2$$ subject to
$$\left\|x\right\|_2 = \left\|y\right\|_2 = 1$$
where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of ...

**0**

votes

**0**answers

20 views

### Kronecker sum matrix for a singular matrix pencil

The Kronecker sum of two matrices $A \in \mathbb{R^{n \times n}}$ and $B \in \mathbb{R^{m \times m}}$ is defined by the matrix.
$$A \oplus B = A \otimes I_m + I_n \otimes B \in \mathbb{R^{mn \...

**0**

votes

**1**answer

72 views

### How eigenvalue perturbation affects back to the original matrix?

Both "eigenvalue perturbation" and "matrix perturbation" seems to study this scenario that given a matrix $A$, if we add something to it like $\tilde{A} = A+E$, how will the eigenvalues of $\tilde{A}$ ...

**7**

votes

**1**answer

137 views

### How should I think about the module of coinvariants of a $G$-module?

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...

**6**

votes

**0**answers

151 views

### A binomial determinant formula: a new variant

In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det_{1\le i,j\le n}\...

**3**

votes

**1**answer

220 views

### Is there any Menelaus-type theorem for polynomials?

Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...

**2**

votes

**1**answer

95 views

### Eigenvalues of A^T D A for positive A and diagonal D

Suppose I have a diagonal matrix $D$ whose entries are bounded in absolute value. I also have a matrix $A$ that is positive (entry-wise, so $A_{ij} > 0\ \forall\ i,j$): one can assume that the ...

**7**

votes

**1**answer

169 views

### The determinant of a $4\times4$ matrix associated to some specific polynomial as follow

Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...

**0**

votes

**0**answers

39 views

### What can we say about the dominant Generalized eigen vector of two matrices A and B if the dominant eigen vector of A and B are known?

I have the following problem: $\bf{a} = V_{max}(A,B)$, where $V_{max}$ refers to the dominant generalized eigen vector solution and A, B are two $N \times N$ matrices (full rank). Suppose i know the ...

**0**

votes

**0**answers

109 views

### Existence of an “almost” skew-symmetric matrix

Let $A\in\mathbb{R}^{3\times 3}$ be a matrix of the form
$$
A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ -a_{12} & a_{22} & a_{23} \\ -a_{13} & -a_{23} & a_{33} \end{bmatrix}
$...

**5**

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**0**answers

139 views

### Can we represent partitions by mutually parallel lines in the plane?

Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...

**-1**

votes

**1**answer

131 views

### Matrix Tic Tac Toe

So we have a 3x3 matrix and two players, a player that only puts in ones and a player that only puts in zeros. A coin flip is used to decide which player goes first. The first move is always to fill ...

**5**

votes

**0**answers

103 views

### A nowhere-zero point in a linear mapping conjecture

I found a very interesting problem in the Open Problem Garden, which I am surprised is not as well-known as I would think it would be:
Prove that If $p>3$ is prime and $A$ is an invertible $n \...

**1**

vote

**1**answer

84 views

### Upper Bounds on the Largest Eigenvalue of Jacobi Matrices

Suppose I have a symmetric tridiagonal (Jacobi) matrix in the following form:
$ \begin{pmatrix}
1 & a_{1} & 0 & ... & 0 \\\
a_{1} & 1 & a_{2} & & ... \\\
0 & a_{...

**0**

votes

**0**answers

38 views

### Is there a formula for computing the parity of a sequence with discrete alphabet?

Suppose we have a sequence of number, whose alphabet is chosen from a discrete set such as (0,1,2). An example of such sequence is 0210122. Now I would like to determine if it is an odd permutation ...

**2**

votes

**0**answers

109 views

### Motivation/intution behind using linear algebra in these combinatorics problems

What is the motivation behind using linear algebra in these three problems ?
A pair $(m,n)$ is called nice if there is a directed graph with (self edge are allowed, but multiple edge are not allowed)...

**8**

votes

**4**answers

646 views

### How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]

Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?

**2**

votes

**0**answers

59 views

### convex approximation for a non convex function

Consider the function
$f\left( {{x_1},...,{x_M},{y_1},...,{y_N}} \right) = \left( {\sum\limits_{j = 1}^M {{\alpha _j}{x_j}} } \right)\left( {{e^{ - \sum\limits_{i = 1}^N {{\beta _i}{y_i}} }}} \right)$...

**2**

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**0**answers

51 views

### Determinant of a rank r perturbation

In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...

**2**

votes

**0**answers

99 views

### Space of change of basis matrices between two similar matrices - how to reduce it with additional tests?

Assume we have two real symmetric $n\times n$ matrices: $A, B$. We can easily test their similarity: $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1..n$. In this case both can be rotated to the same ...

**1**

vote

**2**answers

103 views

### A “positive diagonal plus skew-symmetric” matrix decomposition

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric).
My question. Do there exist an orthogonal ...

**7**

votes

**0**answers

79 views

### Volume of a neighborhood of singular matrices

Suppose we take the set of all $n\times n$ real matrices with entries in $[0,1]$ in Euclidean space. Let $N_\epsilon$ be the $\epsilon$ neighborhood of the set of all singular matrices in this space, ...

**5**

votes

**1**answer

112 views

### On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...

**2**

votes

**0**answers

94 views

### eigenvalues of a square block matrix

How can we show that there are not defective eigenvalues for this square block matrix of dimension $2d \times 2d $: \begin{bmatrix}
A&B\\-B& 0
\end{bmatrix}
where A, B are real matrices, $A =\...

**-3**

votes

**1**answer

86 views

### connected set of sum of upper semi continuous function [closed]

Let $C(X)$:space of continuous functions on a compact space.Topology $C(X)$ is generated by sup-norm($||T||=sup_{v}\frac{||T(v)||}{||v||}$).
Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper ...

**5**

votes

**1**answer

124 views

### Hilbert representation of a bilinear form

Let $\sigma:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be a bilinear symmetric form which is non-degenerate in the sense that for every $0\neq u\in \mathbb{R}^n$ there is $v\in \mathbb{R}^n$ with $...

**2**

votes

**1**answer

52 views

### Separation of two pointed polyhedral cones using hyperplanes generated by facets

Let $C_1$ and $C_2$ two pointed (that is, with vertex in $0$) polyhedral cones in $\mathbb{R}^n$ with $\dim(C_1)=\dim(C_2)=n$. If
$$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\...

**8**

votes

**1**answer

112 views

### Is the lattice generated by finitely many subspaces in a finite-dimensional vector space finite?

Let $V$ be a finite-dimensional vector space, let $U_1,\dots,U_n$ be subspaces, and let $L$ be the lattice they generate; namely, the smallest collection of subspaces containing the $U_i$ and closed ...

**12**

votes

**0**answers

265 views

### Cardinality vs. isomorphism type of vector spaces without choice

One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem:
If $V$ is an infinite vector space over a field $F$, and $...

**0**

votes

**0**answers

29 views

### Vertices of intersection of affine subspace and a cube

Let $A\in\{0,1\}^{n\times n}, b\in\{0,1\}^n$.
Statement: Every vertex $v$ of the polyhedron defined by $\{x\in[0,1]^n|Ax=b\}$ satisfies $v\in\{0,\frac 1 n,\frac 2 n,\dots,\frac {n-1} n, 1\}^n$.
Is ...

**2**

votes

**0**answers

166 views

### A parametrization of stable matrices

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...