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

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0
votes
0answers
6 views

distance from the mean of a normal distribution to the span of a random sample

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from ...
2
votes
2answers
275 views

A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...
3
votes
1answer
48 views

How does grade projection act on homogeneous multivectors in geometric algebra?

I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator. It is noted that $A_r$ and $B_s$ are ...
0
votes
0answers
47 views

Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...
0
votes
0answers
90 views

Checking whether the determinant of a matrix is non-zero [on hold]

For a positive integer $n$, let $c$ be the number of ordered integer tripartitions $(a,b,n-a-b)$ of $n$. Let $F = ( f_{ij} )$ be a $c \times c$ matrix where $f_{ij} = (i+1)^{a + b} (i+2)^{b}$ where $i$...
1
vote
0answers
93 views

Generating $\mathfrak{so}(7)$

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is actually a subalgebra isomorphic to $\...
0
votes
0answers
55 views

Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$. Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is $M[i,j]={(...
-3
votes
0answers
29 views

function mapping odd numbers to counting numbers [closed]

Mapping even numbers to counting numbers is straight forward. Without introducing any other variable: i = 0, 2, 4, 6, . . . if i > 0: count = i/2 what about ...
5
votes
1answer
177 views

Bijection modeling isomorphism of infinite-dimensional vector spaces

Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality. Does there exist a bijection $f : B_V \to B_W$ such that, for each $b_V \in B_V$,...
0
votes
0answers
32 views

$nD$ rotation around a general $(n-2)$-dimensional subspace [closed]

According the Rodrigues' Rotation Formula $3D$ rotation matrix $\in$ $SO(3)$ corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\...
-5
votes
0answers
25 views

How can you prove that f(x) DNE when f(x)+f(2-x)=3x^2+4x+2? [closed]

How can you prove that f(x) DNE when f(x)+f(2-x)=3x^2+4x+2?
-3
votes
0answers
77 views

Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [closed]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...
2
votes
0answers
55 views

How to find a closed form of following matrix's determinant [closed]

I wanna find a closed form of determinant of the following matrix $$A(n) = \begin{pmatrix} B_{1} & B_{2} & \cdots & B_{n} & 1 \\ B_{n} & B_{1} & \cdots & B_{n-1} &...
0
votes
0answers
45 views

Variant of Holder's inequality [migrated]

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
-3
votes
0answers
25 views

A question about linear operator [closed]

Let V be a finite-dimensional vector space over field R. Let $A$ be a linear operator on V. And let $A^3$+ $A$ = $0$. Prove that $tr(A) = 0$.
1
vote
0answers
36 views

Basis for a set of polynomials in Sage? [closed]

I have a large set of polynomials in the coordinates $x,y,z$ in Sage, (e.g. $x^5y-3x^2y^2+2xy^3+x^2yz-y^2z$). I want to know, for example, if $x^5y$ is in the span of my set. Is there a Sage command ...
3
votes
1answer
139 views

The spectral norm of the truncated exponential of a matrix

Let $A$ be a matrix satisfying $A^*+A\leq0$, it can be shown that $\|e^{tA}\|_2\leq1$ for all $t\geq 0$, where $\|\cdot\|_2$ is the spectral norm defined as largest singular value of the matrix. I am ...
3
votes
1answer
311 views

What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...
0
votes
1answer
81 views

Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...
1
vote
0answers
42 views

Determinant formula related to solutions of a second-order recurrence

Let $A$ be the linear map on the space of complex sequences acting as $$(Au)_{n}=u_{n-1}+a_{n}u_{n}+u_{n+1}, \quad n\in\mathbb{Z},$$ where $\{a_{n}\}$ is a fixed sequence. Let $f=f(z)$ and $g=g(z)$ be ...
0
votes
1answer
33 views

Approximate largest eigenvalue of Monodromy matrix

Does anyone know the procedure (or have pseudo code) to approximating the largest eigenvalue of a monodromy matrix? Or even to approximate the monodromy matrix itself? There is no explicit solution ...
4
votes
1answer
74 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
2
votes
0answers
33 views

Why does the objectivity rule out the convexity?

In the famous work "Ball J M. Convexity conditions and existence theorems in nonlinear elasticity[J]. Archive for rational mechanics and Analysis, 1976, 63(4): 337-403", it was mentioned that the ...
1
vote
0answers
28 views

Practical application of envelope theorem for linear programs

Assume that we have solved a (standard) linear program $$ \text{minimize}_{x\in {\mathbb R^n}}\,\, c_0^Tx, \,\,\,\,\, \text{s.t. } A_0x \leq b_0, $$ and would like to know how sensitive is the optimal ...
-1
votes
0answers
27 views

How to calculate the monodromy matrix of the following ODE system [migrated]

I have the following equation: $$ \frac{dw}{dt} = (-V(t)+\frac{1}{\lambda}F(t)) w, $$ with $t>0$ and parameter $\lambda>0$ The matrices $F(t)$ and $V(t)$ take the form $$ F(t)= \left( \begin{...
0
votes
0answers
25 views

homogeneous singular pencil of matrices

I was reading singular pencil of matrices from F.R Gantmachers book(Vol 2, Chap 12), where he deals with the strict equivalence of two pencil, to introduce the concept of infinite elementary divisor ...
0
votes
0answers
80 views

On the transitivity of the action of the unitary group

Let $H$ be a complete inner product space over either real or complex numbers. If $H$ is complete, for two finite sets of vectors $\left\{e_i\right\}_{i\in I}$ and $\left\{f_i\right\}_{i\in I}$ there ...
0
votes
0answers
72 views

Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
1
vote
0answers
51 views

Largest eigenvalue of signed graph

Let us consider a graph where edges can have weight 1 or -1, such a graph is called signed graph. In a signed graph, a cycle is called balanced cycle when product of weights on its edges is positive ...
0
votes
1answer
87 views

Some questions related to the unitary operators

A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product. What is the name of the analogue for the real case? Orthogonal operator ...
1
vote
0answers
32 views

Proof of non-degeneracy of a particular bilinear form [closed]

Given a bilinear form from V x V where V is of rank n and defined by a basis $\{v_1\, ..,v_n \}$, defined by a fixed matrix A which is n x n, and the equation $f(v,w) = \sum_{i}\sum_{j}a_{ij}b_ic_j$ v ...
11
votes
2answers
509 views

Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
3
votes
0answers
25 views

Selecting columns from multiple matrices to form a well-conditioned matrix

Given multiple matrices of the same size, is there a way to select one column from each matrix to form a well-conditioned matrix? For example, given four 4-by-10 matrices A, B, C, D (real, positive, ...
0
votes
0answers
76 views

Orbit intersection in toral automorphisms

Let $f:(\mathbb{R/Z})^2\to (\mathbb{R/Z})^2$ be a hyperbolic toral automorphism induced by a matrix $A\in GL_2(\mathbb{Z})$. Consider a measurable (wrt to the Lebesgue measure $\mu$ on the torus) ...
-3
votes
0answers
59 views

Minors of a Vandermonde matrix [duplicate]

I am working with the $n$ x $n$ Vandermonde matrix where the "$α_i$'s" form the set of integers from 1 to $n$. That is entry $a_{ij}=i^{j−1}$ What I would like to know is if I delete an equal number ...
0
votes
1answer
81 views

How does the rank of $C_i$ change with $i$?

Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ ...
2
votes
2answers
198 views

Linear systems of equations with singular coefficient matrix [closed]

Consider a consistent system of linear equations $Ax=b$. Let's assume for simplicity that $A$ is square $n \times n$. We are looking for an effectively computable approximate solution $\hat{x}$ in the ...
3
votes
0answers
60 views

How many unimodular lattices does it take to fill a cube with high probability?

Consider $C_a$ in $\Bbb Z^n$ a cube of height $a$ at origin in positive coordinates with one corner at origin. Consider the set $M_c$ of all unimodular matrices in $\Bbb Z^{n\times n}$ with each ...
0
votes
0answers
13 views

Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix). With some ...
0
votes
0answers
45 views

SVD alternatives for symmetric matrices

Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices $$ A = E'E $$ Practically this can be done easily using SVD ...
1
vote
0answers
50 views

max min of ratio of quadratic forms

Consider the optimization over two vectors $x$ and $y$ $$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$ for two positive definite matrices $A$ and $B$. This problem can be ...
0
votes
0answers
80 views

Orthogonal Procrustes problem for sub-spaces?

By Orthogonal Procrustes problem I mean given matrix $A$ and $B$ finding a orthogonal matrix $R$ which most closely maps $A$ to $B$, this has a solution as shown in https://en.wikipedia.org/wiki/...
3
votes
1answer
125 views

A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let $f_{...
0
votes
0answers
145 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
5
votes
1answer
260 views

Vector with many non-zero coordinates

Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...
2
votes
2answers
91 views

Behavior of orbits under small perturbations

Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange. Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...
1
vote
0answers
73 views

Showing positive stability of a matrix constructed from a positive matrix

A is a positive nonsingular matrix. Let $s>\rho(A)$. We want to show that $B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right)$ is a positive stable matrix, i.e., all eigenvalues of this matrix ...
6
votes
4answers
523 views

Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago. For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
0
votes
0answers
49 views

a two dimensional integer bijection

I want to try to find a specific two-dimensional linear integer bijection. This is to be used in a double sum rearrangement. It's kind of complicated, but I would really appreciate if anybody has any ...
0
votes
0answers
80 views

How to write a braiding as a matrix?

Let $V$ be the vector representation of $sl_n$. Then $V \otimes V$ is a $U_q(sl_n)$-module. Suppose that a braiding \begin{align} \Psi: V \otimes V \to V \otimes V \end{align} satisfies the ...