# Matrices: eigenvalues, eigenvectors and diagonalisation

> CIE A-Level Further Mathematics · 9231
> Source: https://www.owlsprep.com/study/cie-9231-u2-matrices-eigenvalues-eigenvectors-and-diagonalisation/

This sub-topic covers characteristic equations, eigenvalue/eigenvector calculation, diagonalisability rules, and matrix power computation for 2x2 and 3x3 matrices per CIE 9231 Further Pure 2 requirements.

**Prerequisites:** [Basic matrix operations, determinants and inverse calculation](https://www.owlsprep.com/study/cie-9231-u1-matrices-basics/); [Solving systems of linear homogeneous equations](https://www.owlsprep.com/study/cie-9231-u1-linear-systems/)

## Learning objectives

- Derive and solve the characteristic equation for 2x2 and 3x3 square matrices
- Calculate distinct and repeated eigenvalues and corresponding non-zero eigenvectors
- Verify diagonalisability of a matrix and construct the PDP⁻¹ decomposition
- Apply diagonalisation to efficiently compute high positive integer powers of matrices

## Deriving the Characteristic Equation

The characteristic equation is the core starting point for all eigenvalue work. For any n x n matrix M, we rearrange the defining eigenvector relation $M\mathbf{v} = \lambda \mathbf{v}$ to get $(M - \lambda I)\mathbf{v} = 0$, where I is the n x n identity matrix.

$$\det(M - \lambda I) = 0$$

**Characteristic Polynomial** — The nth degree polynomial in λ obtained by expanding the determinant of M minus λ times the identity matrix.

*Notation:* \det(M - \lambda I)

**Worked example:** Find the characteristic equation for the 3x3 matrix $M = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$

1. First subtract λ times the identity matrix from M:
2. $$M - \lambda I = \begin{pmatrix} 2-\lambda & 1 & 0 \\ 1 & 2-\lambda & 0 \\ 0 & 0 & 3-\lambda \end{pmatrix}$$
3. Expand the determinant along the third row to simplify calculation:
4. $$\det(M-\lambda I) = (3-\lambda) \times \det\begin{pmatrix}2-\lambda & 1 \\ 1 & 2-\lambda\end{pmatrix}$$
5. Simplify the 2x2 determinant and rearrange terms:
6. $$\det(M-\lambda I) = (3-\lambda)\left((2-\lambda)^2 -1\right) = (3-\lambda)(\lambda^2 -4\lambda +3) = -\lambda^3 +7\lambda^2 -15\lambda +9$$
7. Final characteristic equation (multiply both sides by -1 for standard positive leading coefficient):
8. $$\lambda^3 -7\lambda^2 +15\lambda -9 = 0$$

> **tip**
>
> CIE mark schemes award partial marks for correct determinant expansion even if you make an arithmetic error later, so show every step of the expansion clearly.

**Check your understanding**

1. What is the degree of the characteristic polynomial for a 4x4 matrix?

   *Why:* The degree of the characteristic polynomial always equals the size of the square matrix.

> **Exam tip:** Always expand the determinant along a row or column with zero entries to reduce arithmetic errors, especially for 3x3 matrices.

## Calculating Eigenvalues and Eigenvectors

Once you have the characteristic equation, factorise it to find all roots, which are the eigenvalues. For each eigenvalue λ, substitute back into $(M - \lambda I)\mathbf{v} = 0$ and solve the homogeneous system to find the corresponding non-zero eigenvector.

**Worked example:** Find all eigenvalues and corresponding eigenvectors for the matrix M from the previous example, $M = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$

1. Factorise the characteristic equation we derived earlier:
2. $$\lambda^3 -7\lambda^2 +15\lambda -9 = (\lambda-3)^2(\lambda-1) = 0$$
3. Eigenvalues are λ=1 and λ=3 (repeated twice)
4. For λ=1, substitute into $(M - I)\mathbf{v} = 0$:
5. $$M-I = \begin{pmatrix}1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2\end{pmatrix} \implies v_1 + v_2 = 0, v_3=0 \implies \mathbf{v} = k\begin{pmatrix}1 \\ -1 \\ 0\end{pmatrix}, k\neq 0$$
6. For λ=3, substitute into $(M - 3I)\mathbf{v} = 0$:
7. $$M-3I = \begin{pmatrix}-1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 0\end{pmatrix} \implies -v_1 + v_2 = 0 \implies \mathbf{v} = k\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix} + m\begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix}, k,m \neq 0$$

**Exam command terms**

CIE uses specific command terms for this section:

- **Find the eigenvector** — Any non-zero scalar multiple of the solution is acceptable *(You do not need to normalise the vector to unit length)*

- **Find a set of linearly independent eigenvectors** — You must provide as many distinct vectors as the dimension of the matrix

## Diagonalisability and PDP⁻¹ Decomposition

An n x n matrix is diagonalisable if and only if it has n linearly independent eigenvectors. If this condition holds, you can write $M = PDP^{-1}$, where P is the modal matrix of eigenvectors and D is the diagonal matrix of corresponding eigenvalues.

**Worked example:** Construct the PDP⁻¹ decomposition for the 3x3 matrix M from earlier

1. We have 3 linearly independent eigenvectors, so M is diagonalisable. Arrange eigenvectors as columns of P, matching order of eigenvalues in D:
2. $$P = \begin{pmatrix} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}$$
3. Calculate the inverse of P using standard 3x3 inverse method:
4. $$P^{-1} = \frac{1}{2}\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$
5. Verify the decomposition by multiplying PDP⁻¹ to confirm you recover M.

## Calculating High Matrix Powers Using Diagonalisation

A key application of diagonalisation is computing $M^n$ for very large n, which would be impossible via repeated matrix multiplication. The identity $M^n = PD^nP^{-1}$ holds, and $D^n$ is trivial to compute as you just raise each diagonal entry to the power n.

**Worked example:** Compute $M^4$ for our example 3x3 matrix M

1. Raise each diagonal entry of D to the 4th power:
2. $$D^4 = \begin{pmatrix} 1^4 & 0 & 0 \\ 0 & 3^4 & 0 \\ 0 & 0 & 3^4 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 81 & 0 \\ 0 & 0 & 81 \end{pmatrix}$$
3. Multiply P, D⁴ and P⁻¹ in sequence:
4. $$M^4 = \begin{pmatrix} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 81 & 0 \\ 0 & 0 & 81 \end{pmatrix} \frac{1}{2}\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix} = \begin{pmatrix}41 & 40 & 0 \\ 40 & 41 & 0 \\ 0 & 0 & 81\end{pmatrix}$$

## Common pitfalls

- **Wrong:** Normalising eigenvectors to unit length unnecessarily
  - Why it fails: Wastes time and introduces arithmetic errors that lose marks, as CIE does not require unit eigenvectors
  - Correct: Leave eigenvectors in simplest integer component form with no fractions or radicals
- **Wrong:** Mixing up the order of eigenvectors in P and eigenvalues in D
  - Why it fails: The columns of P must exactly match the order of eigenvalues on the diagonal of D, otherwise the decomposition is invalid
  - Correct: Label each eigenvector with its corresponding eigenvalue before building P and D to confirm alignment
- **Wrong:** Using the zero vector as an eigenvector
  - Why it fails: The zero vector is explicitly excluded from the definition of eigenvectors, and will not satisfy the non-trivial solution condition
  - Correct: Always specify that the scalar multiple constant k is non-zero when writing general eigenvector solutions
- **Wrong:** Assuming all matrices with repeated eigenvalues are not diagonalisable
  - Why it fails: Repeated eigenvalues can still have a full set of linearly independent eigenvectors, making the matrix diagonalisable
  - Correct: Count the number of linearly independent eigenvectors before concluding a matrix is non-diagonalisable
- **Wrong:** Forgetting to multiply by P⁻¹ at the end when calculating Mⁿ
  - Why it fails: This is the single most common mark-losing error in CIE diagonalisation questions, worth 2-3 marks
  - Correct: Write the full formula $M^n = PD^nP^{-1}$ at the top of your working to remind yourself of all three terms

## Cheatsheet

| Step | 2x2 Matrix | 3x3 Matrix | CIE Mark Scheme Check |
| --- | --- | --- | --- |
| 1. Characteristic Equation | Expand 2x2 determinant, solve quadratic | Expand along zero row/column, solve cubic | Confirm sum of eigenvalues equals trace of M |
| 2. Eigenvectors | Solve 1 linear equation for non-zero solution | Solve homogeneous system for each λ | No zero vectors, any non-zero multiple accepted |
| 3. Diagonalisation | 2 independent eigenvectors required | 3 independent eigenvectors required | P and D order must match exactly |
| 4. Mⁿ Calculation | Raise D entries to n, multiply PDⁿP⁻¹ | Raise D entries to n, multiply PDⁿP⁻¹ | Verify with small n (e.g. n=2) to catch errors |

## What's next

Mastering eigenvalues, eigenvectors and diagonalisation is the foundation for the rest of the Further Pure 2 matrices unit. You will next extend these concepts to symmetric matrices, which have the unique property that their eigenvectors are mutually orthogonal, allowing you to construct orthogonal diagonalisation. This is a heavily weighted 15-20 mark question type in recent CIE exam papers. You will also go on to learn the Cayley-Hamilton theorem, which lets you compute matrix powers and inverses without full diagonalisation, and explore applications of matrix transformations to conic sections. These concepts build directly on the skills you have practiced here, so ensure you can complete all 3x3 diagonalisation steps without arithmetic errors before moving on.

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