A square matrix is called diagonal if all entries not on the main diagonal are zero.

We will show that if $$A$$ has $$n$$ distinct eigenvalues, then the computation of $$A^k$$ can be replaced by $$D^k$$, where $$D$$ is a diagonal matrix with the eigenvalues of $$A$$ as entries.

### Matrix Summary of Eigenvalues of $$A$$

Let $$A$$ be an $$n \times n$$ matrix and let $$\lambda_1, \lambda_2, …., \lambda_n$$ be scalars and $$\vec v_1, \vec v_2, … , \vec v_n$$ be nonzero vectors in n-space. Let $$C$$ be the $$n \times n$$ matrix having $$\vec v_j$$ as $$j$$th column vector, and let $$D$$ be the diagonal matrix with entries of all lambda scalars.

Then $$AC = CD$$ if and only if all $$\lambda$$ are eigenvalues of $$A$$ and $$\vec v_j$$ is an eigenvector of $$A$$ corresponding to $$\lambda_j$$

$$C$$ is invertible if and only if $$rank(C) = n$$. That case, we can write the equation as $$D = C^{-1}AC$$.

### Diagonalizable Matrix

An $$n \times n$$ matrix $$A$$ is diagonalizable if there exists an invertible matrix $$C$$ such that $$C^{-1}AC = D$$, a diagonal matrix.