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

learn more… | top users | synonyms

4
votes
2answers
68 views

Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?

I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...
9
votes
1answer
112 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...
5
votes
3answers
142 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 ...
1
vote
1answer
213 views

Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$. What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product ...
0
votes
1answer
200 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
1
vote
0answers
138 views

Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$ Let me give a ...
7
votes
1answer
96 views

Compute only selected components of an eigenvector

I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...
4
votes
1answer
152 views

$n$ columns of a specific “infinite” Vandermonde matrix always linearly independent? [on hold]

Consider the "infinite" Vandermonde matrix $$ V (x_1, x_2, \ldots , x_n) = \begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\ ...
0
votes
0answers
28 views

basis span of space necessary to be orthogonal? [on hold]

If a vector space V that span of {v1,v2,....,vk},can the basis vector of V are not mutually orthogonal? For vector space V, are there exist set of orthogonal basis {o1,o2,....,ok} that can span V? ...
0
votes
0answers
53 views

Elementary Linear Algebra Maximization Problem [on hold]

Can someone show me the proof for the following: min $\frac{q^TLq}{q^TWq}$ where q is not 0 and subject to $qWe =0$ is solved when q is the eigenvector corresponding to the 2nd smallest eigenvalue ...
-1
votes
0answers
60 views

Why solving a system of linear equation produces the intersection of the equation [closed]

Let us consider two equations 1)x+y=1 2)-x+y=1 Consider the solution of the equations using Gaussian Elimination, Geometrically I am able to understand the ...
3
votes
1answer
93 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
9
votes
2answers
1k views

Questions on Toeplitz Matrices

These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in MathStackExchange. Let $A$ be an $n\times n$ Hermitian Toeplitz matrix: $$ A = ...
0
votes
0answers
34 views

Derivative of softmax loss function [closed]

I am trying to wrap my head around backpropagation in a neural network with a softmax classifier, which uses the softmax function: \begin{equation} p_j = \frac{e^o_j}{\sum_k e^{o_k}} \end{equation} ...
0
votes
1answer
38 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 ...
-1
votes
1answer
73 views

Action of rotation group on Matrices [closed]

Is the following assertion true? Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in ...
-3
votes
1answer
117 views

A problem that involves matrix and Lorentz Transformation [closed]

To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes. $1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G ...
5
votes
1answer
2k views

How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?

I have a matrix $K$ which is the sum of a identity and a Kronecker product of two symmetric matrices as following and I want to calculate the inverse of it $K^{-1}$. \begin{eqnarray} ...
-2
votes
0answers
60 views

How to plot 2-D and 3-D Joint numerical range of real symmetric matrices in mathematica? [closed]

I know this question this doesn't belong here. However, I am not getting a satisfactory reply from the mathematica forum. Consider $2\times 2$ real symmetric matrices $\mathbf{A}_1$ and ...
1
vote
1answer
67 views

Connection between eigenvalues of A and its LDL decomposition

Consider an undirected graph $G$ with $N$ vertices and its adjacency matrix $n_{ij}$: $n_{ij} = 1$ if vertices $i$ and $j$ are connected by an edge and $n_{ij} = 0$ otherwise. Consider $A_{ij} \equiv ...
1
vote
0answers
40 views

Convergence to Eigenvalue gap of Gaussian Orthogonal Ensemble

Let $M_n$ come from the Gaussian Orthogonal Ensemble of size $n\times n$. Let $E_1(0; I)$ be the probability that $M_n$ has no eigenvalues in an interval $I$. The bulk scaling limit of this is defined ...
3
votes
1answer
252 views

Number of Matrices with bounded determinant

Here's my question: Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
0
votes
0answers
110 views

Is $Rank(A)=Rank(A^{T})$ where $A$ has inifinite rows and columns and its given that $A$ has finite rank [closed]

Is $Rank(A)=Rank(A^{T})$ where $A$ has inifinite rows and columns and its given that $A$ has finite rank. Is it a sub-case of finite rank operators which maps finite dimensional compact sets(which ...
1
vote
1answer
156 views

Diagonalization of 4th order tensors

I have been wondering about the following problem... Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...
10
votes
2answers
377 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
1
vote
2answers
100 views

The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...
3
votes
0answers
52 views

Dimension of the sum of images of transpose

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\im}{\operatorname{im}}$ Given $A,B\in M_{n\times n}(k)$, define $\rank(A,B):=\dim(\im A+\im B)$. I'm looking for results regarding relationships ...
19
votes
0answers
616 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
9
votes
1answer
211 views

What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
4
votes
2answers
275 views

Variety determined by interior product of the determinant?

Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., ...
2
votes
1answer
56 views

Solution to generalized Sylvester equation

I am interested in solving generalized Sylvester equations (for $X$) of the form: $$ \sum_{j=1}^k A_j X B_j^T = F, $$ where $A_j,B_j,X,F\in\mathbb{C}^{n\times n}$ and $k$, $n$ are integers. I will ...
8
votes
2answers
763 views

How to transform matrix to this form by unitary transformation?

Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e. $$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$ Then is ...
0
votes
2answers
111 views

Proof for a Rank-One Decomposition Theorem of Positive (semi) Definite Matrices

Consider the following result which I recently came across in a research paper in my area (Signal Processing) Let $X$ be a $N\times N$ positive semidefinite (psd) matrix whose rank is $r$. Let ...
5
votes
0answers
187 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
1
vote
0answers
76 views

Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that $$Tx = \begin{pmatrix}A & B \\ C & D ...
1
vote
1answer
157 views

Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...
0
votes
1answer
89 views

Symmetric Zero-Diagonal Matrices

Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1_n\\1_n&0\end{array}\right]$. Then $X^t\Omega X$ is ...
0
votes
1answer
95 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 ...
0
votes
0answers
56 views

Speed up Linear programming

I have a linear programming problem like this: minimize $c^t X$ under the constraint that $AX \ge b$. I will need to solve this linear programming problem online many times. I need it to be as fast ...
4
votes
1answer
118 views

standard practice for large dense truncated svd computations?

What are the standard methods of computing the rank-k truncated SVD of large dense matrices? My literature search yields results only for large sparse matrices. I assume for k small that you use a ...
1
vote
1answer
94 views

Checking the intersection of two sets

Let $E\subset{\mathbb R}^n$ be a set of the type $I_1\times \dots \times I_n$, where $I_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is ...
0
votes
0answers
25 views

Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
1
vote
0answers
73 views

Bounds on the effect of a matrix product on the Frobenius norm

I was wondering if there was a way to put upper and lower bounds on the Frobenius norm of a matrix product in relation to the Frobenios norm of one of the individual matrices, i.e, ...
4
votes
1answer
160 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 ...
0
votes
0answers
34 views

Multiplicative Symmetrization of Bilinear Forms

Let $F$ be a field of characteristic $2$, let $V$ be an even dimensional $F$-vector space, let $B$ be a non-degenerate symplectic bilinear form on $V$ and let $^*$ be the adjoint of $B$ on ${\rm ...
0
votes
0answers
45 views

Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases. My intention is to: decompose $\Phi$ via SVD: $U \Lambda V^T = \Phi$ to create it's ...
1
vote
0answers
64 views

A criteria for a subalgebra of M(n,C) being M(n,C) [migrated]

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...
-2
votes
0answers
26 views

non-negativity of cost function

For which set of real parameters $\alpha $ and $\beta$ the following expression is negative and for which positive $\frac{1}{\alpha \beta} \sum_{i=1}^n \log \frac{\alpha \lambda_i^{\beta}+\beta ...
9
votes
0answers
189 views

Determinant inequality involving Hermitian, positive definite matrices

Question: Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question ...
9
votes
1answer
251 views

Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity

I'm currently trying to get familiar with the Jordan normal form for matrices; and after some example I ask for the possible Jordan-form for the Carleman matrix for the function $f(x) = \sin(x)$ when ...