# Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

## Archive for Januar, 2010

In [1, p. 34 exercise 2.11] we are asked to find the eigenvalues of the circulant matrix given in [1, (2.27),p.22]. read the conclusion >
In [1, p. 34 exercise 2.10] it is asked to prove that if $\mathbf{A}$ is a complex $n \times n$ positive definite matrix and $\mathbf{B}$ is a full rank complex $m \times n$ matrix with $m \leq n$, then $\mathbf{BA}\mathbf{B}^H$ is also positive definite. read the conclusion >
In [1, p. 34 exercise 2.9] we are asked to prove that the rank of the complex $n\times n$ matrix
 $\mathbf{A}=\sum\limits_{i=1}^{m} d_{i}\mathbf{u}_i\mathbf{u}_{i}^{H}$ (1)

where the $\mathbf{u}_i$ are linearly independent complex $n \times 1$ vectors and the $d_i$‘s are real and positive, is equal to $m$ if $m \leq n$. Furthermore we are asked what the rank equals to if $m$ > $n$. read the conclusion >