Eigendecomposition

Let \( A \) be an \( n \times n \) matrix with \( n \) linearly independent eigenvectors \( v_1 ... v_n \), then we can write \( A \) as:

\[ A = PDP^{-1} \]

Where \( P \) is the matrix whose ith column is \( v_i \) and \( D \) is a diagonal matrix with \( D_{i,i} = \lambda_i \).

Proof:

\begin{array}{r l l} &Av_i &= \lambda_iv_i \\ \implies& AP &= PD \\ \implies& A &= PDP^{-1} \end{array}