Diagonalisation

Diagonalisation is the process of expressing a matrix in the form $A = PDP^{-1}$A=PDP−1, where $D$D is a diagonal matrix of eigenvalues and $P$P is a matrix of eigenvectors. This is extremely useful for computing matrix powers and understanding the long-term behaviour of repeated transformations.

---

The Diagonalisation Formula

If an $n \times n$n×n matrix $A$A has $n$n linearly independent eigenvectors, then:

$A = PDP^{-1}$A=PDP−1

where:

• $D = \begin{pmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{pmatrix}$D=​λ1​0⋮0​0λ2​⋮0​⋯⋯⋱⋯​00⋮λn​​​ is the diagonal matrix of eigenvalues
• $P = \begin{pmatrix} \mathbf{v}_1 & \mathbf{v}_2 & \cdots & \mathbf{v}_n \end{pmatrix}$P=(v1​​v2​​⋯​vn​​) is the matrix whose columns are the corresponding eigenvectors

---

When Is a Matrix Diagonalisable?

A matrix is diagonalisable if and only if it has n linearly independent eigenvectors (where $n$n is the matrix size).

This is guaranteed when:

• The matrix has $n$n distinct eigenvalues
• The matrix is symmetric (always diagonalisable with real eigenvalues)

It may fail when:

• There are repeated eigenvalues and insufficient independent eigenvectors

---

2×2 Worked Example