problems4.2 - Math 307: Problems for section 4.2 November...

Math 307: Problems for section 4.2
November 14, 2012
1. (i) What can you say about the diagonal elements of a Hermition matrix?
(ii) Show that if A is an n × n matrix such that ( v ,A w ) = ( A v , w ) then A is Hermitian.
2. Show that if A is any matrix then A * A and AA * are Hermitian with non-negative eigen-
values.
3. Follow the procedure in the notes to find an orthogonal matrix V such that V
T
AV is
upper triangular when A =
bracketleftbigg
1 2
1 1
bracketrightbigg
.
4. Explain why the Laplacian matrix L for a resistor network has non-negative eigenvalues.