Matrix diagonalization

Given a matrix with linearly independent eigenvectors , we can form a matrix whose columns are these eigenvectors. We can likewise form a diagonal matrix whose diagonals are given as the eigenvalues corresponding to each . This matrix is the diagonalization of .

We can show that is similar to :

1. Since for each and , it follows that
2. Since is a matrix of linearly independent eigenvectors, we know that it can be inverted. Multiplying both sides by , we obtain .

Hence for any matrix with linearly independent eigenvectors, we know that is similar to .