This article is mainly notes I have taken from reading Linear Algebra (Addison-Wesley, 1995) section 5.2. For course MATA22 at UTSC.

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.