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

learn more… | top users | synonyms

6
votes
2answers
163 views

What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $n \times n$ matrix ...
8
votes
1answer
181 views

Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute \begin{equation} I = \det \left(\mathbf{I} + \mathbf{A}\right) \end{equation} where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers: $$ \left\{ ...
0
votes
1answer
57 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C. I'd like a function μ:Cn×n→[0,∞) ...
-1
votes
0answers
40 views

Tensor and Hom are bi-adjoint functors for finite dimensional vector spaces? [on hold]

Let $U$, $V$ and $W$ be finite dimensional vector spaces over a field $\mathbb{F}$. It is well known that $U\otimes_\mathbb{F}-$ and $\mathrm{Hom}_\mathbb{F}(U,-)$ are adjoint functors in the sense ...
0
votes
0answers
43 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
-11
votes
0answers
63 views

In Bedrock the citizens can rent cars from Wreck Rental for $40 down and $0.04 per mile. [on hold]

But they can also rent from Cutie Pie Rental for $0.20 per mile. After how many miles does Wreck Rental become a better deal?
-9
votes
0answers
46 views

The ordered pair (3,100) indicated that it costs $100 for a job that is 3 hours work. He charges $190 for 13 hours of work [on hold]

Find the slope between these two points. Slope = _____________ Find the y-intercept __________________ Write a linear equation to represent this scenario. Where x is the number of hours and y is the ...
2
votes
1answer
360 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 ...
0
votes
0answers
48 views

inequality involving determinants and quadratic forms [on hold]

I'm interested in comparing $\det(x'\boldsymbol{A}x)$ and $\det(\boldsymbol{A})x'x$ where $\boldsymbol{A}$ is symmetric positive semidefinite, and $x$ is a free vector of constant. My argument is: ...
4
votes
2answers
230 views

“Typical” convergence rate for the von Neumann mean ergodic theorem

The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$, where ${\cal H}$ is a Hilbert space, the following holds: $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
2
votes
1answer
171 views

Generalizing the spectral radius of a unistochastic matrix

Consider a square matrix $A$, and from it construct $B$ whose entries are the squared magnitudes of those in $A$. What can we say about the spectral radius of $B$? I know that for a unitary matrix ...
1
vote
1answer
31 views

Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
-4
votes
0answers
32 views

I have to show any non-invertible matrix is a reducible matrix [on hold]

Suppose that $A \in M_n(D)$ and $D$ be a division ring. An $n × n$ matrix $A = (a_{ij} )$ is called reducible if $A$ has a non-trivial invariant subspace in $D^n$. I have to show any non-invertible ...
1
vote
1answer
54 views

Phase of the inner product between the elements of an ETF

I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question: Is there any research about the phase of inner ...
52
votes
3answers
9k views

Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is: A vector space has the same dimension as its dual if and only if it is finite dimensional. I have seen a total of one proof ...
4
votes
3answers
952 views

Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?

Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
-1
votes
0answers
43 views

bounding of the largest Singular value of a product of matrices [closed]

Does anybody know the inequality of the largest singular value for product of matrices, i.e. ?? > $\sigma \uparrow (XY)$ > ??, X and Y are positive definite matrices. The bounds should be linear so ...
6
votes
2answers
312 views

Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
7
votes
3answers
3k views

Solving a system of linear inequalities — what is the dimension of the solution set?

It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$? For the applications I have in mind the ...
0
votes
0answers
10 views

How to treat non-identifiable states in Kalman filtering/dynamic linear models?

Let $x_t = G_tx_{t-1}+\omega_t$ with $\omega_t \sim \mathrm{N}(\mathbf{0}, \mathbf{W}_t)$ be a state equation and $y_t = F_tx_t+\nu_t$ with $\nu_t \sim \mathrm{N}(\mathbf{0}, \mathbf{V}_t)$ be a ...
1
vote
0answers
97 views

Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: ...
26
votes
16answers
7k views

Linear Algebra Proofs in Combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
0
votes
4answers
140 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
5
votes
2answers
198 views

Minimum of squared sum minus sum of squares

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates. I'm ...
8
votes
0answers
390 views

On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots, f_k(x) \in \mathbb F_q[x]$ be ...
1
vote
0answers
44 views

non-intersecting families of subspaces

Given $V$, a vector space over a finite field $F$ of size $k$, if $\dim(V)=m$, and $r$ divides $m$, there exists a family of $r$-dimensional subspaces, whose size is equal to $(k^m-1)/(k^r-1)$ and ...
1
vote
1answer
109 views

Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...
6
votes
1answer
179 views

Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$. A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
1
vote
2answers
78 views

Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation: $\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$ which is essentiall in matrix notation: $\dot{\mathbf{x}} = ...
0
votes
0answers
61 views

Existing complete function space under suitable norm

This question was asked in math.stackexchange.com but no suitable answer was received, so I am posting it here. This is a question which came to me due to several previous question: sorry for the all ...
1
vote
0answers
51 views

Smallest sum of original column entries in 2d matrix [closed]

I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this: ...
1
vote
0answers
74 views

What is the trace of this operator in $L^\infty$ (if this question make sense)?

Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator : ...
5
votes
1answer
295 views

Finding a basis for the (linear combinations) span of a matrix group, efficiently?

I have an algorithm whose bottleneck is the following task: Let $\mathbb{F}$ be a finite field. Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let $G=\langle ...
5
votes
1answer
318 views

How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field. Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra. Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ that one needs to take ...
1
vote
1answer
38 views

Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$, we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$ and $T$ be sets of elements of $\mathbb{F}$ of size ...
1
vote
1answer
53 views

Given $M$, minimize $|Mx|_0$

Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find $\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$, where the $\ell_0$ "norm" is measured by simply counting the number ...
1
vote
0answers
48 views

Question about Eigenvalues of group elements [migrated]

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
2
votes
1answer
170 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 ...
9
votes
0answers
185 views

Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
4
votes
3answers
775 views

Tensor product of linear mappings versus chain complexes

A chain complex of vector spaces $X_k$ is a sequence of linear mappings $\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} ...
1
vote
1answer
1k views

Does this linear algebra construction based on a graph have a name, and where has it been studied?

In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used : Question : Have such constructions been used elsewhere, and if so ...
0
votes
1answer
228 views

Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
8
votes
1answer
267 views

Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
2
votes
0answers
294 views

How to prove the following determinant identity? [migrated]

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
0
votes
0answers
41 views

Issue of Tikhonov regularization and Sobolev spaces

This question related to the link below: Shttp://math.stackexchange.com/questions/1272235/space-of-tikhonov-regularization-of-an-ill-poised-problems Tikhonov regularization gives smoothing results ...
1
vote
0answers
152 views

Is there a way to simplify this apparently huge characteristic polynomial calculation?

Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let ...
0
votes
0answers
62 views

Degree of permutation of hypercube

Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
2
votes
1answer
135 views

Linear independence of +/- 1 strings/vectors II

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...
6
votes
4answers
421 views

Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
5
votes
1answer
209 views

Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...