# Linear algebra: orthonormal basis

1. May 5, 2013

### Felafel

1. The problem statement, all variables and given/known data
$\phi$ is an endomorphism in $\mathbb{E}^3$ associated to the matrix
(1 0 0)
(0 2 0) =$M_{\phi}^{B,B}$=
(0 0 3)

where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1))

Find an orthonormal basis "C" in $\mathbb{E}^3$ formed by eigenvectors of $\phi$

3. The attempt at a solution

Being the eigenvalues the elements of the diagonal 1, 2, 3
Aren't (1, 0, 0), (0,2,0), (0,0,3) three orthonormal vectors already?

Or should I write the endomorphism according to the canonical basis first and find new values?

2. May 5, 2013

### Staff: Mentor

Yes, they are, but these aren't the vectors they're asking for.
Use the eigenvalues to find a basis of eigenvectors, and then make an orthonormal basis out of that set of vectors.