---
title: "Diagonalisation and Powers of a Matrix"
description: "When enough independent eigenvectors exist, the map is a diagonal matrix in the eigenbasis, which makes powers, long-run behaviour and matrix exponentials easy."
canonical: https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX
source: https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX.md
retrieved: 2026-09-12
---

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# Diagonalisation and Powers of a Matrix

When enough independent eigenvectors exist, the map is a diagonal matrix in the eigenbasis, which makes powers, long-run behaviour and matrix exponentials easy.

Subject: Mathematics · Area: Linear Algebra · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX

## Ready when they can

- Diagonalise a matrix and state the two factors involved
- Compute a high power of a matrix through its diagonal form
- Give a matrix that cannot be diagonalised and say what fails

## Lesson: Eigenbases make powers easy

Diagonalising writes A as P D P inverse, where D holds eigenvalues on its diagonal and P holds matching eigenvectors as columns. The recipe: find all eigenvalues, find a basis of eigenvectors, stack them into P, read D off. A diagonal matrix shows it plainly: diag(4, 7) has eigenvalues 4 and 7.

**Example.** Powers then collapse beautifully: A to the k equals P D to the k P inverse, with D to the k just raising diagonal entries. With D = diag(2, 3), A squared has eigenvalues 4 and 9, so its trace is 13. With eigenvalues 1 and 2, A to the 10th has trace 1 + 1024 = 1025.

This works exactly when enough independent eigenvectors exist to fill P. Distinct eigenvalues always supply them. But [[2, 1], [0, 2]] has eigenvalue 2 twice with only one eigen direction, so no eigenbasis exists and it cannot diagonalise.

**Tip.** An eigenspace is just the null space of A minus lambda I, so eigenvector hunting reuses your null space skills. After assembling, multiply out P D P inverse once to confirm you land back on A.

**Recap.** Stack eigenvectors into P, eigenvalues into D, and let the diagonal do the heavy lifting for powers.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Dimension and the Rank-Nullity Theorem](https://lightmysky.com/learn/mathematics/dimension-and-the-rank-nullity-theorem-mt_aMG8PcwPAR)
- [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa)

## Opens up

- [Modules: Linear Algebra Over a Ring](https://lightmysky.com/learn/mathematics/modules-linear-algebra-over-a-ring-mt_0GMVp-s8wD)
- [Systems of Differential Equations and Eigenvalue Solutions](https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur)
- [Stationary Distributions and Long-Run Behaviour](https://lightmysky.com/learn/mathematics/stationary-distributions-and-long-run-behaviour-mt_MGfHvUGkeh)
- [Inner Products, Length and Orthogonality](https://lightmysky.com/learn/mathematics/inner-products-length-and-orthogonality-mt_X0HPRGto4W)