0
votes
0answers
30 views

Bounding Rayleigh quotioent for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
0
votes
1answer
57 views

dual space of the quotient space of some locally convex topological space

I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
4
votes
0answers
75 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
17
votes
2answers
698 views

Example of a space for which $V \cong Hom(V,V)$

Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology. Is there a non-trivial ...
1
vote
1answer
157 views

A linear algebraic q-difference equation [SOLVED]

I would like to solve the following algebraic linear q-difference equation: \begin{equation} a\left(x\right)f\left(x\right)=f\left(qx\right) \end{equation} The parameter $q$ is real, positive and ...
2
votes
0answers
59 views

System of 2 linear q-difference equations with singular matrix

I would like to solve the following algebraic linear system of q-difference functional equations: \begin{cases} ...
4
votes
1answer
88 views

Weak ergodicity of nonhomogenous products of 0-1 matrices

Here is a question which probably has a negative answer, but I couldn't find any literature directly on it. Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
7
votes
1answer
83 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 ...
0
votes
0answers
63 views

Interpret some coefficients in algebras

Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...
4
votes
1answer
462 views

A question on Grassmannian

Let $V$ be the space of $4$ by $4$ Hermitian matrices, that is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform measure of $$ \left\{ W\in Gr\left(5,V\right):W \text{contains no ...
0
votes
0answers
154 views

Closed-form expressions for dual norms of real normed vector spaces

Didn't get any biters over at MSE, so I figure this place might be more appropriate... Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by ...
2
votes
1answer
112 views

Existence of an “Orthogonalizing” Operator

I was wondering if it was possible to prove existence of a unitary operator $A$ such that: $\langle Au,u\rangle=0$ for all $u$. In 2-dimensions it clearly is (just a 90 degrees rotation) and similarly ...
4
votes
1answer
146 views

Are dual spaces barreled?

Let $X$ denote a topological affine space (with no additional assumptions). Let $X^*$ denote its dual space of continuous affine functionals, equipped with the weak-$*$ topology. It is easy to see ...
4
votes
1answer
241 views

Isomorphisms between topological vector spaces [closed]

Let $f : X \to Y$ be a continuous map of complete topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A ...
4
votes
1answer
330 views

Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space

I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask. First, consider the following form ...
3
votes
0answers
66 views

Algebraic conditions of separability

Let $X$ be a real vector space (without any norm), and $Y$ be a convex subset of $X$, $0\notin Y$. The goal is to find a hyperplane $L$ passing through 0 such that $Y$ lies in a closed halfspace ...
0
votes
0answers
32 views

Last Point of Exit from the 2-D Positive Quadrant

I define the functions $f_i(\mathbf{u}),i=1,2$ and $g_i(\mathbf{u}),i=1,2$ where $\mathbf{u}\in\mathbb{C}^{N}$ is the unit-norm vector. Thus, this functions are defined over the unit norm $N-$sphere. ...
2
votes
2answers
199 views

Existence of a projection operator onto a classical set of density matrices

I have a Hilbert space of quantum density matrices written in the Glauber-Sudarshan P representation - ie. we have coherent states $|\alpha \rangle$ and we write density matrices as $$ \rho = \int ...
3
votes
1answer
154 views

Schur product, partial order

Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner partial order. How to ...
2
votes
1answer
175 views

Coercive Symmetric Bilinear form on a Hilbert space

I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance. 1) Consider a continuous symmetric ...
2
votes
1answer
202 views

lipschitz constant of a multivariate function

I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...
0
votes
1answer
235 views

All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k

The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in ...
3
votes
0answers
148 views

Extending a Hilbert space isometrically

Let $H$ be a Hilbert space, and let $X$ be a topological vector space. Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$? ...
2
votes
2answers
376 views

Relationship between the derivative of a matrix and its eigenvalues

Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing? ...
4
votes
2answers
443 views

When is spectral norm of AB equal to that of BA?

I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$ The argument I was using to show this is that for any two square ...
2
votes
1answer
162 views

Factorization of bivariate polynomial

Let $q(y, z) = u_1 + u_2y + u_3 z + u_4y^2 + u_5yz + u_6z^2 + u_7y^3 + u_8y^2z + u_9yz^2 +$ $\hspace{2.55cm}u_{10}y^3z + u_{11}y^2z^2 + u_{12}y^3z^2$ Can $q(y, z)$ be factorized as ...
8
votes
2answers
336 views

Rescaling positive definite matrices to force a unit eigenvector

Hello, Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones. I'm hoping to construct a positive, diagonal matrix $W$ such that $$(W X'X W) \mathbf{1} = ...
2
votes
1answer
235 views

Decomposing bilinear forms in Hilbert spaces

You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...
8
votes
3answers
419 views

Relating a Polynomial equation to the characteristic equation of a Hermitian matrix

This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
0
votes
0answers
239 views

Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of ...
0
votes
1answer
137 views

Positive definite Hermitian matrices of countable rank

Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the ...
2
votes
0answers
123 views

Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$

Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range. That ...
5
votes
0answers
120 views

reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then $$ \|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2). $$ ...
0
votes
0answers
164 views

The ratio of two strictly increasing functions

Given: \begin{equation} f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i \end{equation} \begin{equation} f_2(a)=\sum_{i=1}^{k^*-1} ...
7
votes
2answers
330 views

polynomials with minimal $L_\infty$ norm on multiple disjoint intervals

It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
22
votes
16answers
2k views

functions satisfying “one-one iff onto”

Hello Everybody. I need some more examples for the following really interesting phenomenon: A function from the class ... is one-one iff it is onto. Some ...
4
votes
1answer
251 views

Spectral Properties of $A(I-A)^{-1}$

I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
1
vote
1answer
184 views

references for families of conditionaly negative definite matrices

We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have $$ ...
11
votes
1answer
701 views

Decomposition of positive definite matrices.

It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum $$ A=\sum_{j} B_j \otimes C_j $$ with $B_j$ and $C_j$ positive semidefinite matrices (of ...
10
votes
1answer
859 views

Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite. My question is: Is there a ...
0
votes
1answer
118 views

Maximal length vector under constraints

Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
5
votes
1answer
356 views

positive hermitian elements in $M_n(\mathbb{C})$

Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers have some special properties: (i) they are closed under sum, (ii) they are closed under multiplication by positive ...
5
votes
0answers
213 views

Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
0
votes
2answers
334 views

Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$

A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space ...
3
votes
2answers
789 views

trace norm inequality for positive matrices

If $A, B$ are positive $n \times n$ complex matrices, $n$ some integer, then obviously \begin{equation*} \|ABA\|_\text{tr} = tr(ABA) = tr(A^2 B). \end{equation*} But can we say there is a constant ...
1
vote
1answer
1k views

Dual operators between Hilbert spaces : With or without riesz representation

Let $X$ and $Y$ be Hilbert spaces over the real numbers (so complex conjugation plays no role, and everything will be linear in the strict sense). Let $f : X \rightarrow Y$ be a linear continuous ...
0
votes
1answer
287 views

When are operators extended by linearity bounded?

Greetings. Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
6
votes
1answer
437 views

Spectra of a Symmetric Toeplitz Operator

For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form $\begin{bmatrix} ...
0
votes
1answer
423 views

Extending Continuous Sublinear maps on dense subsets of a Banach space

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying: (1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||. Please prove ...
2
votes
0answers
423 views

Eigenvector of infinite matrix

I consider the system of reaction-diffusion PDEs in a ball with Robin boundary condition. It is a Steklov eigenvalue problem (see G Auchmuty (2004) "Steklov eigenproblems and the representation of ...