2
votes
1answer
64 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 ...
0
votes
1answer
34 views

What does left composition mean in this question?

Consider the vector space of all linear transformations $L(V,V)$ on the vector space $(V,K)$ and a linear map $F:L(V,V)\to L(V,V)$ such that $F(a)= b \circ a$ for all $a\in L(V,V)$, where $b\in ...
2
votes
1answer
60 views

Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$.

I want to Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$. Where $A$ is a matrix. I have a proof that involves Jordan Blocks. The proof is long and involved but it not ...
2
votes
1answer
287 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
5
votes
1answer
160 views

Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform ...
4
votes
2answers
209 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 ...
1
vote
1answer
134 views

How to find the unknown values in this Numerical Integration type?

Given the following type of numerical integration: $$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$ a) Find the values ​​of: the coefficient $c_1$ and points $x_0$ and $x_1$ so ...
2
votes
1answer
132 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
vote
1answer
90 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
1answer
97 views

Matrix inversion of an analytical function

Following problem: I have a function $f(x_1,x_2)$ and Im looking for the inverse $finv(x_1,x_2)$ of the function which is defined through: $\int f(x_1,y)\cdot finv(y,x_2) d y =\delta(x_1,x_2) $ ...