Linear algebra: orthonormal basis

[Felafel]

Homework Statement

$\phi$ 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 $\phi$

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?

 

[Mark44]

Homework Statement

$\phi$ 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 $\phi$

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?

Yes, they are, but these aren't the vectors they're asking for.

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

Use the eigenvalues to find a basis of eigenvectors, and then make an orthonormal basis out of that set of vectors.