No! The theorem you stated is not an if and only if! It only says, IF the eigenvalues are distinct, THEN the matrix is diagonalizable. And not the other way round.

You may read

http://en.wikipedia.org/wiki/Symmetric_matrix

there you can find among other things:

"Every real symmetric matrix A can be diagonalized"

No! The theorem you stated is not an if and only if! It only says, IF the eigenvalues are distinct, THEN the matrix is diagonalizable. And not the other way round.

You may read

http://en.wikipedia.org/wiki/Symmetric_matrix

there you can find among other things:

"Every real symmetric matrix A can be diagonalized"