0
votes
0answers
17 views

Possible to find inverse or eigenvalues of a block diagonal matrix with upper and lower diagonal matrices

I just encountered a matrix problem of finding inverse of eigenvalues of a block diagonal matrix with upper and lower also matrices of the form where A and B are full rank matrices. Is there any ...
1
vote
3answers
121 views

Proof for diagonalizable matrix

Let $A \in M_n(\mathbb C)$ be invertible. Prove that $A$ is diagonalizable if and only if $A^{-1}$ is diagonalizable. This is what I have for one direction of the proof: Suppose $A$ is ...
1
vote
1answer
37 views

Updating eigen decomposition for a matrix after some row changes

Let us say we have a matrix $A$ which has eigen decomposition $$A=UDU^{-1}$$ If some of the rows of A are changed by multiplying a constant positive value, is there a simple way to update the eigen ...
4
votes
2answers
207 views

nonegative inverse eigenvalue problem

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\ ...
3
votes
1answer
74 views

How to compute $\text{trace}((A+D)^{-1}A)$

Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute $$\text{trace}((A+D)^{-1}A)$$ Or is there a good approximation?
0
votes
1answer
285 views

Eigenvalues of a Toeplitz matrix

A toeplitz matrix $X$ is given as, \begin{equation} X = \begin{pmatrix} ~~~~\textbf{1} ~~~~\textbf{c} ~~~~\textbf{d} ~~~~0 ~~~~0 ~~~~0 ~~~~\textbf{d}~~~~ \\ ~~~~\textbf{c} ~~~~\textbf{1} ...
2
votes
1answer
496 views

Calculate a whitening matrix without using inverses?

Consider a random column vector $\mathbf{x}$, of dimension $m$. That is, it is a random vector, composed of $m$ random variables. The PDF of the random vector $\mathbf{x}$ is thus the joint-PDF of its ...
0
votes
1answer
484 views

Matrix elements of an inverse Hermitian matrix

I am trying to tackle the following exercise in a quantum chemistry textbook: Show that: If $\mathbf{G}(\omega) = (\omega \mathbf{1}-\mathbf{A})^{-1}$, and $\mathbf{A}$ is Hermitian (i.e. ...