-1
votes
0answers
15 views

prove that F is dense in C(X×Y,R)?

Let X,Y be compact metric spaces. Let F= {∑ Ai fi(x) gi(y),fi∈∁(X,R),gi∈∁(Y,R), i from 1 to n }. prove that F is dense in C(X×Y,R) ? please i cant figure it out any help i will be thankful !
2
votes
1answer
52 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
1
vote
2answers
31 views

set of symmetric positive definite matrix open?

I consider a collection of symmetric positive definite matrices of the same dimension. I've learned it's an open set but have no clue about the proof. Also, can the symmetry condition be dropped? ...
0
votes
1answer
18 views

Operator norm of real matrix

I've been looking through my workbook in preparation for the next set of classes and I'm stuck on this problem and don't know how to possibly proceed with it. The hint isn't helping and there isn't ...
0
votes
5answers
120 views

Do non-square infinite matrices exist?

Sorry, I tried to wrap my head more around this, but I failed. Given non-square matrix $A$ that has dimension $kn \times n$. Now let $n$ goto infinity. Is the matrix finally square?
1
vote
1answer
23 views

Is it true that $|(Mv)\cdot(Nw) \leq C|v||w|$ (matrix-vector)?

If $M$ and $N$ are matrices such that each element of the matrices depends on $t$, so we have $M_{ij}(t)$, $N_{ij}(t)$, and we have the result $M_{ij}$, $N_{ij} \in L^\infty(0,T)$, is it true that ...
2
votes
1answer
97 views

Linearly independent functionals

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
0
votes
0answers
32 views

question about a invertibility of a matrix

Let $Q_n$ a finite dimentional space of a Hilbert space $(Q,(\cdot,\cdot)_Q)$ and let $b$ a bounded bilinear form $b:Q_n\times Q_n\to \mathbb{R}$ such that it's elliptic, i.e., there exists $c>0$ ...
0
votes
1answer
49 views

A matrix in $SL(2)$ has it's supremum norm and infimum fulfilled by orthogonal vectors

I am having trouble proving the next statement: If $B\in SL(2)$ and $||B||\neq 1$, for $||B||:= \underset{x\neq 0}{\sup}\big\{\frac{||B(x)||}{||x||} \big\} $, where $||\cdot||$ is the euclidian norm, ...
0
votes
1answer
88 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
1
vote
1answer
25 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
0
votes
3answers
68 views

Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
1
vote
1answer
48 views

Relation of norms of matrices

Let $A$ be $m \times n$ matrix. Let $B=\frac 1n AA^*$, where $A^*$ is a transposed matrix. Let $X_i, I\leq m$ be row-vectors of $A$. Show $$ \|B\|=\frac 1n \|A\|^2\geq \max_{i\leq m}|X_i|, $$ Where, ...
4
votes
0answers
78 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
1
vote
1answer
28 views

$|Av||A^{-1}v|$, $A$ non-singular, $|v|=1$.

Let $A$ a non-singular $n \times n$ matrix, $v \in \mathbb{R}^{n}$ a variable vector. The operator norm of $A$ is defined to be $|| A ||=\max_{|v|=1} |Av|$ where $|Av|$ is the standard Euclidean norm ...
1
vote
2answers
45 views

Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$
1
vote
2answers
167 views

Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in ...
2
votes
2answers
65 views

Maximum of two positive operators

Let $A,B$ be two positive operators in $B(H)$. Does there exist, in general, an operator $C$ such that for each $T$, if $A \leq T$ and $B \leq T$, then $$A\leq C \leq T\quad \text{and}\quad B\leq ...
0
votes
1answer
42 views

characterization of an infinite matrix mapping and continuity

Show that an infinite matrix mapping $A=[a_{ij}]$ $:l^{\infty}\to l^{\infty}$ is continuous iff $sup_{i\in \mathbb N}$ $ \sum_{k=1}^{\infty}{|a_{ij}|}=||A||<\infty$. Give a characterization of the ...
2
votes
2answers
149 views
3
votes
1answer
320 views

Eigenvalues of matrix with entries that are continuous functions

For each $t \in [0,b]$, let $M(t)$ be an $n \times n$ matrix with entries $m_{ij}(t).$ The matrix $M(t)$ is invertible and positive-definite, so the eigenvalues of $M(t)$ exist and are positive for ...
1
vote
0answers
43 views

How to show that entries of this matrix are in $L^\infty(0,T)$?

I have a problem. Let $A(t)$ be a $n \times n$ matrix for each $t \in [0,b]$ with the property for all vectors $x$ that $$x^TA(t)x \geq C|x|^2$$ where $C$ doesn't depend on $t$. Can I use this fact ...
1
vote
0answers
40 views

Matrix with elements in $L^\infty(0,T)$ means inverse matrix also has elements in $L^\infty(0,T)$?

Let $A(t)$ be a matrix with entries $a_{ij} \in L^\infty(0,T)$. Suppose $A(t)^{-1}$ exists, and is positive-definite. Does it follow that $A(t)^{-1}$ has elements in $L^\infty(0,T)$ too?
3
votes
0answers
46 views

Supremum over unitary group action

Let $A$ and $B$ are two given Hermitian operators on matrix algebra $M_n(\mathbb{C})$. $A$ is positive semi-definite with unit trace. I want to know the general method for calculating the following ...
1
vote
0answers
112 views

Iwasawa Decomposition

I was asked to prove that if $$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$ denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...
2
votes
1answer
126 views

Adjoint matrix eigenvalues and eigenvectors

I just wanted to make sure that the following statement is true: Let $A$ be a normal matrix with eigenvalues $\lambda_1,...,\lambda_n$ and eigenvectors $v_1,...,v_n$. Then $A^*$ has the same ...
4
votes
2answers
165 views

Is Householder orthogonalization/QR practicable for non-Euclidean inner products?

The question Is there a variant of the Householder QR algorithm to orthonormalize a set of vectors with respect to an inner product if no orthonormal basis is known a priori? Background Let's ...
3
votes
1answer
117 views

Norm of a linear transformation

Let $T:\mathbb R^2\to \mathbb R^2$ be given by the matrix $\begin{pmatrix}a&b\\ c& d\end{pmatrix}$. Let $u:=a^2+b^2+c^2+d^2+2(ad-bc)$ and $v:=a^2+b^2+c^2+d^2-2(ad-bc)$. I need to show that ...
2
votes
0answers
42 views

Meaning of nonlinear vectorial equation

I am trying to apply some methods in a paper and I have to solve the following fixed point equation from Proposition VIII.4.3 in Asmussen (2000): $$\mu_+ =\mu ...
2
votes
1answer
45 views

limit of evaluated automorphisms in a Banach algebra

Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible ...
3
votes
1answer
114 views

Skew symmetric matrix decomposes

I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $ R $ such that $ R^T A R = M$, ...
2
votes
1answer
182 views

Uniform convergence and interchange or sum and integral in Cauchy integral formula

I am working with the Cauchy Integral Formula for a matrix $A$ over a closed contour $C$. I have the following calculation, I believe this is correct, but I don't understand why I am allowed to ...
8
votes
1answer
75 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
1
vote
0answers
33 views

What is the matrix norm in defining the generator of a continuous time Markov chain?

For a continuous time Markov chain with finite state space and Markov transition function $p(t)$, its generator $G$ can be defined entry-wise as $$ G_{i,j}:=\lim_{t\to 0^+} \frac{p_{i,j}(t) - ...
0
votes
0answers
78 views

All matrix/vector norms induce the same topology?

From Wikipedia all norms on $K^{m \times n}$ are equivalent; they induce the same topology on $K^{m \times n}$. This is true because the vector space $K^{m \times n}$ has the finite dimension $m ...
1
vote
1answer
55 views

Matrix completion: supplementary questions

Continuation of the question here, what is going to happen if we change the some of the conditions. I write it as a quote from here and change the appropriate places which are underlined: I need ...
2
votes
1answer
122 views

Relation between positive definite Hermitian matrices with their inverses

Let $A$ and $B$ be two positive definite Hermitian matrices. Show that the Hermitian matrix $$C\ =\ A^{-1} + B^{-1} - 4(A + B)^{-1}$$ is also positive definite. Thanks in advance.
-1
votes
2answers
69 views

Prove $f(A^T)=f(A)^T$ for a matrix $A$ [closed]

As the title says, I need to prove $f(A^T)=f(A)^T$ for a matrix $A$. (where $T$ is the transpose) I believe the proof involved the fact that an interpolation polynomial $r(A)=f(A)$ and then I must ...
1
vote
1answer
76 views

Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
2
votes
0answers
41 views

Are decomposable maps completely bounded?

By the word decomposable I mean a positive map $\phi:\mathcal{B(H)}\rightarrow \mathcal{B(K)}$; $\mathcal{H,K~}$ are some Hilbert spaces and $\phi=\psi_1+T\circ \psi_2$ where $T$ is the transpose ...
1
vote
1answer
117 views

Matrix completion

I need to find an algorithm (if exists) of the following matrix completion problem. I need to construct $n^2$ positive semi-definite matrices, say $\{P_i\}_{i=1}^n$. Entries of these matrices are ...
2
votes
3answers
65 views

Solving for positive semidefiniteness

Given a real matrix M, is there a matrix function f(M) such that $f(M)-M$ is guaranteed to be positive semidefinite, other than the idea of multiplying $M$ with its transpose and apart from the ...
6
votes
1answer
206 views

Cauchy Integral Formula for Matrices

How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
1
vote
2answers
81 views

Using Neumann series to compute $T^{-1}$

Need help on how to show that $S$ satisfies the necessary condition for Neumann series. Here is what is given. $T\in B(X,X)$ where $X$ is a Banach space. Let $T: \mathbb R^3 \rightarrow \mathbb R^3$ ...
1
vote
0answers
48 views

2 positive decomposable maps

A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
7
votes
0answers
192 views

Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
1
vote
1answer
114 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
0
votes
1answer
106 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
2
votes
1answer
113 views

Convergence in norm independent of the choice of the norm.

When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing $\left\Vert X_n - Y_n\right\Vert\rightarrow 0$ as $n\rightarrow 0$ ...
2
votes
3answers
91 views

norm of a product of matrices

Suppose for every $x \in \mathbb{R}$ and $y \in [0,1]$, $M(x,y)$ is an $n$ by $n$ matrix and suppose that for every $y \in [0,1]$, $M(x,y) \to M_\infty$ as $|x| \to \infty$, where $M_\infty$ is a ...