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?

Quote:

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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]

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Let z = x + iy and solve for x.

Quote:

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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]

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Apply a well known theorem about the eigenvalues of and .

Quote:

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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?

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Read this: http://www.math.umn.edu/~garrett/m/algebra/notes/24.pdf