Suppose that the eigenvalues of are distinct. Then the corresponding eigenvectors are linearly independent.
We prove it by contradiction. For this we assume that are linearly dependent. Then we can find the largest integer so that are linearly independent, but are not. Thus, we should be able to find a nontrivial solution to the homogeneous equation
As a corollary, we can find that is diagonalizable.
EXAMPLE 2. Determine whether the matrix is diagonalizable or not.
EXERCISE 3. Diagonalize each of the following matrices if possible.