Homework Help: Linear algebra - Spectral decompositions: Eigenvectors of projections

1. Apr 19, 2010

TorcidaS

1. The problem statement, all variables and given/known data
Let P1 and P2 be the projections defined on R^3 by:

P1(x1, x2, x3) = (1/2(x1+x3), x2, 1/2(x1+x3))
P2(x1, x2, x3) = (1/2(x1-x3), 0, 1/2(-x1+x3))

a) Let T = 5P1 - 2P2 and determine if T is diagonalizable.
b) State the eigenvalues and associated eigenvectors of T.

2. Relevant equations

3. The attempt at a solution

For a), I believe it is diagonalizable because P1 + P2 gives us (x1, x2, x3). Although I'm could be wrong on that...

It's mainly b) that I'm concerned for. By the theorem, (T = c1P1 + c2P2...+.. where c are eigenvalues) 5 and -2 are the eigenvalues (although this sort of confuses me because I had thought the eigenvalues of projections are always 1 and 0).

How can we find the eigenvectors? Had this been a matrix it's simple, subtract the eigenvalue from the main diagonal, simplifiy, and find the nullspace.

Also, the solution to b) is for eigenvalue 5, the eigenvectors are <(1,0,1),(0,1,0)> and for eigenvalue -2, the eigenvector is <(-1,0,1)> (so my guess some matrix is formed?)