Diagonalization Theorem and related theorems

Diagonalization Theorem

$A$ is diagonalizable iff eigenvectors of $A$ are a basis for $R^n$.

Given that A is diagonalizable, the jth column of P will contain the jth vector from the basis of eigenvectors, and the jth column of the diagonal matrix D will contain the corresponding eigenvalue in the (j, j)−entry

See examples in lecture notes.

Corollary 1

$A$ is diagonalizable iff $a_{\lambda }=g_{\lambda }$ for every $\lambda$ of $A$.

Corollary 2

If $A_{n\times n}$ has $n$ eigenvalues, then $A$ is diagonalizable.