Orthogonal Matrices and Symmetric Matrices

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!

Re: Orthogonal Matrices and Symmetric Matrices

why not do this:

for i ≥ j define B = (b_{ij}) by b_{ij} = a_{ij} and for i < j, b_{ij} = a_{ji}.

clearly B is symmetric.

now let C = A - B. on or below the diagonal (i ≥ j), c_{ij} = a_{ij} - b_{ij} = 0.

thus C is upper triangular with all 0's on the diagonal.

example:

$\displaystyle A = \begin{bmatrix}1&4&7\\-5&3&2\\1&0&3 \end{bmatrix}$.

then:

$\displaystyle B = \begin{bmatrix}1&-5&1\\-5&3&0\\1&0&3 \end{bmatrix};\ C = \begin{bmatrix}0&9&6\\0&0&2\\0&0&0 \end{bmatrix}$.

it seems to me we can do this no matter what the eigenvalues of A are.

for 2: another way to write |x| is as the scalar:

√(x^{T}x). thus |Px| = √((Px)^{T}Px) = √(x^{T}(P^{T}P)x)

what is P^{T}P?

Re: Orthogonal Matrices and Symmetric Matrices

Thank you very much Deveno, your suggestion for #2 was great and I managed to figure out 'a' and 'b' from it!

I am still, however, confused about #1, am I not losing full generality when I define B? And am I "allowed" to do such a thing? Sorry if this feels very fundamental - but I've not taken a class on formal math proofs or logic.

And thanks again in general. You've helped me tons and I don't believe I've said that yet.

Re: Orthogonal Matrices and Symmetric Matrices

how? we are given a matrix A, we are forming B by considering "half" of A, and "making up the difference" with C.

Re: Orthogonal Matrices and Symmetric Matrices

Ah, thank you. I get it now.