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

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4
votes
0answers
88 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
0
votes
1answer
117 views

Dimension of Commutator Space [closed]

For each $n\times n$ matrix $A$ with real entries the set $$C(A)=\{X\in M_n(\mathbb{R}): AX=XA\}$$ is obviously a linear subspace of $M_n(\mathbb{R})$. Can we recognize the dimension of this ...
0
votes
1answer
135 views

Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define: \begin{align*} U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\ V &= {\rm diag} \{ \frac{1}{\alpha_i} ...
2
votes
0answers
69 views

Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent \begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation} of a 3-tensor $W_{j,k,l}$ of ...
0
votes
1answer
57 views

Adjusting matrix in generalized eigenvalue problem for the design of eigenfunctions

Take two matrices $A,B \in \mathbb{R}^{n\times n}$. I am considering the generalized eigenvalue problem ($\gamma \in \mathbb{R}, \phi \in \mathbb{R}^n$) $$A\phi = \gamma B\phi.$$ Is there a ...
7
votes
1answer
240 views

Block Matrix determinant

Consider the $k \times k$ block matrix: $ C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \ddots & ...
2
votes
1answer
98 views

How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$ $k(x, x') = (1 - 2|x-x'|)$ is a positive semidefinite function? It turns out to be psd function when ...
3
votes
1answer
114 views

$\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$. Moreover, let $M \colon \mathbb{R}^n \to ...
2
votes
0answers
74 views

Separating duality for TVS?

What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ? ...
0
votes
0answers
65 views

Bounding multiplications of PSD random matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
9
votes
2answers
356 views

Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
2
votes
1answer
353 views

Who defined and who coined “module”?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether? EDIT   In view of @anon's and KConrad's answers, and as it could have been ...
4
votes
1answer
49 views

Eigendecomposition of analytic Hermitian matrix-valued functions of several variables

If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda_i(t)$ and $x_i(t)$ corresponding to the eigenvalues and ...
2
votes
1answer
303 views

Operator norm vs spectral radius for positive matrices

I believe the following statement should be true but somehow I don't see an argument: For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
2
votes
1answer
71 views

Find base of kernel with as many 0 as possible

I have a 400x132 rectangular matrix with only 0 and 1. I am looking for the linear combinations of the columns of the matrix that sum to 0. For example C1 + C2 - C3 = 0. I want to find the linear ...
2
votes
0answers
66 views

Linear transformation of a polyhedral cone

Let $C$ be a polyhedral cone in $\mathbb{R}^m$ with H- and V-representations $C = \{x : A x \le 0 \} = \{R y : y \ge 0\}.$ The pair $(A,R)$ is referred to as a double description (DD) pair of the ...
3
votes
1answer
106 views

Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers? In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...
4
votes
2answers
208 views

Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint"). I know that all the eigenvalues of matrices in this ...
0
votes
0answers
7 views

Complex parameters in the Ritz procedure

I am using the Ritz procedure to write a trial function as the superposition of other admissible functions, with the coefficients being unknown variational parameters to be determined. The variational ...
0
votes
1answer
77 views

Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
8
votes
1answer
444 views

Reference request: Book of Linear algebra from categorical point of view

Is there any book of Linear algebra in the modern language of Category theory? I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...
4
votes
2answers
121 views

When does a cone contain its dual cone?

Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in ...
0
votes
1answer
67 views

Efficient way to find SVD of sum of projection matrices?

Lets say that we have n matrices of data $X_i : i \in [1, n]$. All $X_i$ have the same number of rows. Their associated projection matrices are $P_i = X_i(X_i^T X_i)^{-1}X_i^T$ Also say that we have ...
4
votes
1answer
358 views

Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?

Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
2
votes
1answer
420 views

An inequality involving traces and matrix inversions

The following question kept me wondering for some time: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
1
vote
1answer
106 views

Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let \begin{equation*} p = p\left(\kappa,d\right) := ...
6
votes
2answers
228 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ Here, $H$ is an NxN skew-Hermitian matrix (for very ...
1
vote
0answers
28 views

Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
2
votes
1answer
71 views

Is first term of my cost function convex?

I have an optimization problem in the form of [\begin{array}{l} \mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...
0
votes
1answer
60 views

Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows \begin{eqnarray*} ||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...
3
votes
1answer
132 views

Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators. We want to look at matrices that agree in most of their entries and ...
3
votes
1answer
80 views

On the solution of a generalized Lyapunov equation

We shall reconsider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is a positive integer and $C$ is a known symmetric positive semidefinite matrix. I met with this ...
3
votes
1answer
100 views

weak version of a Baez-Crans 2-vector space?

Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces ...
0
votes
0answers
60 views

Relation between the block maximum norm and the Euclidean norm of a matrix

I am trying to give answer to the following question: Let's define de block maximum norm of a $N*M \times N*M$ matrix as \begin{eqnarray*} \parallel A \parallel_b = max_{x \neq 0} [ \parallel A x ...
0
votes
0answers
96 views

Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
4
votes
1answer
438 views

Die hard nilpotent spaces

Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is nilpotent if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned ...
2
votes
2answers
142 views

Matrix, singular values, Moore-Penrose-pseudoinverse

If A is any real mxn-matrix consider the block matrix $\begin{pmatrix} E&A^T \\ A&0\end{pmatrix}$. This matrix seems to have close connections with pseudo inverse, svd etc. which are probably ...
0
votes
0answers
48 views

Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
0
votes
1answer
62 views

Integral over Kronecker product

Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that $$ \int_{0}^T A(t) dt $$ is invertible. Does it then follow that the matrix $$ \int_{0}^T A(t) \otimes ...
4
votes
1answer
76 views

Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...
10
votes
1answer
207 views

Factor a sum of products of cofactors

Let $M$ be any $n\times n$ matrix. We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$. We can write ...
7
votes
3answers
379 views

Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...
1
vote
0answers
99 views

Default Orientation of Vectors [closed]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...
6
votes
1answer
196 views

Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$ i.e. Horn's inequalities for n matrices? Can such ...
2
votes
0answers
48 views

Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...
9
votes
1answer
280 views

Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$. Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$: $$R_q = M_q^T (M_q ...
5
votes
3answers
159 views

Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e. $$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$ It is known that for arbitrary $A_{n\times n}$ $$\|T\circ ...
2
votes
2answers
107 views

Boundedness of ratio of linear functions

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
1answer
143 views

Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...
3
votes
1answer
143 views

What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...