Hi, I have two problems that I'm really struggling with:

1) If A is an n x n matrix with real eigenvalues, show that A = B + C where A is symmetric and C is nilpotent. Hint: Use the fact that an n x n upper triangular matrix U with zeroes on the main diagonal satisfies U^n = 0.

What I think/know:

I know that there exists a P, such that P inverse = P transpose (P^T) where P^(T)AP = U ; where U is an upper triangular matrix. I tried taking a power of n of the equation:

A^n = (PUP^(T)) = (A + B)^n , but that doesn't seem to get me anywhere. Please show me how else I can approach the problem and a kick start!

2) Let P denote an n x n matrix.

a) If P is orthogonal, Show |Px| = |x| for every column x in R^n.

b) If |Px| = |x|, show that P is orthogonal.

What I think/know:

I know that since P is orthogonal, the rows and columns of P are orthonormal. I tried multiplying them out as P = [ C1 C2...Cn ] and x = [xi]; but when I took the norm of it, I couldn't get anywhere. Perhaps my block multiplication is wrong - I have a hard time wrapping my head around "seeing" 'n' terms of the columns of P times the column of x. Help please! Also - I assumed that part b) would be similar and maybe once I got a), I could try b) alone. If it seems like I assumed incorrectly, please help as well!