Are all the following statements true?
Any subspace of R^n has an orthogonal basis.
Every square matrix is diagonalizable.
0 vector is an eigenvector of every square matrix.
If B=P^(-1)AP then A and B have the same eigenvalues.
Every linear transformation has an eigenvalue.
In this thread you will find 2 or 3 matrices that aren't diagonalizable.