Characteristic roots of Hermitian and skew-Hermitian matrices

[Hala91]

Homework Statement

1)Prove that the characteristic roots of a hermitian matrix are real.
2)prove that the characteristic roots of a skew hermitian matrix are either pure imaginary or equal to zero.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

What is the definition of "Hermitian matrix"? Have you worked with "self-adjoint linear operators" yet?

 

[Hala91]

A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose - that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j
NO we haven't worked with it yet...

 

[Hala91]

Honestly I have no clue how to prove any of them :S

 

[lanedance]

for the first one, start by considering an eigenvalue of H
$$Hu = \lambda u$$

or similarly consider the characteristic equation
$$| H- \lambda I|$$

consider the hermitian conjugate of either arguments

 

[Hala91]

Thanks for your help guys I have proved them earlier this morning :)