Diagonalizing Matrices via Eigenvectors

Factoring a matrix into its eigenbasis decouples coupled linear systems into simple scalar scalings.

Eigenbasis Factorization

An n-by-n matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. Stacking these vectors as columns forms an invertible change-of-basis matrix P that translates standard coordinates into coordinates aligned with the transformation's invariant axes.

A = P D P^{-1}

The Diagonal Matrix

The matrix D is diagonal and holds the eigenvalues corresponding to each column of P in matching order. Operating in this basis strips away all cross-coupling between variables, reducing the linear transformation to independent one-dimensional stretches.

D = \operatorname{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)

Accelerating Matrix Powers

Directly computing powers of a dense matrix requires repeated matrix multiplications. Because intermediate factors of P and its inverse cancel telescopically in the diagonalized form, raising A to any power requires exponentiating only the scalar diagonal entries.

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

Flashcard

P^{-1} A P = D

Similarity transform projecting a matrix into diagonal form

Conjugating a matrix by its modal matrix of eigenvectors changes the coordinate system to the eigenbasis, where the operator acts purely as coordinate-wise scaling.

Check your understanding

Given a diagonalizable matrix A with decomposition A = P D P^{-1}, which expression correctly computes A^3?

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