# Matrices / Complex Numbers /eigenvalues+vectors

• March 29th 2010, 05:22 AM
thatgirlrocks
Matrices / Complex Numbers /eigenvalues+vectors
I really need some help answering this question, it's doing my head in!!

Q:

a) let z in the set of C (complex numbers). show that if z+zbar = 0, then Re(z) = 0 (something to do with Argand diagrams?)

(where zbar is z with a horizontal line across the top :S)

b) let A in the set of R^nxn (a square matrix, n by n) be skew-symmetric, that is Atransposed = -A. show that all eigenvalues of A have zero real-part.

c) suppose that x and y are eigenvectors corresponding to distinct eigenvalues of a real-values skew-symmetric matrix. show that (xbartransposed)y=0

can anyone help me solve this please?
• March 29th 2010, 02:25 PM
mr fantastic
Quote:

Originally Posted by thatgirlrocks
I really need some help answering this question, it's doing my head in!!

Q:

a) let z in the set of C (complex numbers). show that if z+zbar = 0, then Re(z) = 0 (something to do with Argand diagrams?)

(where zbar is z with a horizontal line across the top :S)

[snip]

Let z = x + iy and solve for x.

Quote:

Originally Posted by thatgirlrocks
[snip]
b) let A in the set of R^nxn (a square matrix, n by n) be skew-symmetric, that is Atransposed = -A. show that all eigenvalues of A have zero real-part.
[snip]

Apply a well known theorem about the eigenvalues of $A$ and $A^T$.

Quote:

Originally Posted by thatgirlrocks
[snip]
c) suppose that x and y are eigenvectors corresponding to distinct eigenvalues of a real-values skew-symmetric matrix. show that (xbartransposed)y=0

can anyone help me solve this please?