why not do this:
for i ≥ j define B = (bij) by bij = aij and for i < j, bij = aji.
clearly B is symmetric.
now let C = A - B. on or below the diagonal (i ≥ j), cij = aij - bij = 0.
thus C is upper triangular with all 0's on the diagonal.
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:
√(xTx). thus |Px| = √((Px)TPx) = √(xT(PTP)x)
what is PTP?