Diagonalization

Japanese school year: University year 1

What you learn

Learn how to transform square matrices into diagonal matrices using eigenvectors and invertible matrices. Diagonalization is extremely useful for computing matrix powers and solving systems of differential equations. Prior mastery of finding eigenvalues, eigenvectors, and matrix inverses is necessary.

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Key points

Use this formula to simplify square matrix $A$ into a diagonal matrix with eigenvalues $\lambda_1$ to $\lambda_n$ along its diagonal. Matrix $P$ is formed by arranging the eigenvectors.

P^{-1} A P = \operatorname{diag}(\lambda_1, \dots, \lambda_n)

Use this formula to easily calculate powers of matrix $A$ ( $A^k$ ). Raising diagonal matrix $D$ to power $k$ is simple, as you only raise each diagonal number to the power $k$ .

A^k = P D^k P^{-1}

This formula shows how to construct matrix $P$ for diagonalization. Arrange eigenvectors $\boldsymbol{v}_1$ to $\boldsymbol{v}_n$ as columns in the same order as eigenvalues in the diagonal matrix.

P = [\boldsymbol{v}_1 \ \dots \ \boldsymbol{v}_n]

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Practice 1Coming soon