Symmetric matrix problem

Question

$A$ is a symmetric matrix and has a eigenvalue $\lambda$ of order $m$.

Why $\lambda$ has $m$ independent eigenvectors?

I want to prove $A$ is diagonalizable by proving it has $n$ independent eigenvectors. And I know that different eigenvalue has independent eigenvectors but I don't have an idea about the same eigenvalues.

Answer 1

I hereby introduce you the Schur decomposition, which is a simple decomposition can be used to prove the diagonalization of symmetric matrix very quickly.

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

After decomposing $A$ into the upper triangular form, you can prove that it is indeed the diagonal form.