Eigenvalues and eigenvectors
An eigenvector keeps its direction under a linear transformation, up to sign and scale.
An eigenpair satisfies Av = λv with v nonzero. Rearrange to (A − λI)v = 0. A nonzero solution is possible when det(A − λI) = 0, giving the characteristic equation for λ.
After finding an eigenvalue, solve the homogeneous system for a corresponding nonzero vector. Any nonzero scalar multiple is also an eigenvector. Repeated eigenvalues need care: their algebraic multiplicity need not equal the number of independent eigenvectors. Edexcel places this in FP2; AQA includes it in compulsory pure.
Worked example
Find the eigenvalues of [[2,1],[0,3]].
- det(A − λI) = (2 − λ)(3 − λ).
- Set the determinant equal to zero.
- The two factors give the two eigenvalues.
Answer: λ = 2 or λ = 3
Revise first: Determinants and singular matrices.
What survives a matrix transformation?
An eigenvector is a nonzero vector whose image is a scalar multiple of itself: Av = λv. The eigenvalue λ gives the scale and sign along that direction. The zero vector satisfies the equation for every λ, which is why it cannot be an eigenvector.
For a 2 × 2 matrix, det(A − λI) = 0 is a quadratic. After finding λ, solve (A − λI)v = 0 and check both rows. A negative eigenvalue reverses direction; zero sends the eigenvector to the origin.
If you can express an initial vector as a combination of independent eigenvectors, repeated transformations become powers of the eigenvalues. This shortcut needs a full eigenvector basis; a repeated eigenvalue alone does not guarantee one.
Repeatedly applying [3, 1; 1, 3]
- The characteristic equation is (3 − λ)² − 1 = 0, so λ = 4 or 2.
- For λ = 4, (1, 1) is an eigenvector. For λ = 2, (1, −1) is an eigenvector.
- Write (3, 1) = 2(1, 1) + (1, −1).
- After two transformations the image is 2 × 4²(1, 1) + 2²(1, −1) = (36, 28).
- Direct multiplication gives A(3, 1) = (10, 6) and A(10, 6) = (36, 28).
Is (0, 0) an eigenvector? Does det A = 0 mean every vector maps to zero?
No to both. A singular matrix has at least one nonzero vector in its kernel, but may have many nonzero images.
Take the idea further
For a repeated eigenvalue, test whether two independent eigenvectors actually exist before using a diagonalisation argument.
- matrices similarity eigenvalues
- matrices eigenvector ratio
- matrices power action
- matrices singular parameter
Practise eigenvalues and eigenvectors
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 9FM0 · FP2 · 3.1 Further matrix algebra: Eigenvalues of a 2 × 2 matrix
- H245 · additional-pure · Matrices: Eigenvalues of a 2 × 2 matrix
- H645 · Y435 · Matrices: Eigenvalues of a 2 × 2 matrix
- 7367 · core · C10: Eigenvalues of a 2 × 2 matrix
- 9FM0 · FP2 · 3.1 Further matrix algebra: An eigenvector of a 2 × 2 matrix
- H245 · additional-pure · Matrices: An eigenvector of a 2 × 2 matrix
- H645 · Y435 · Matrices: An eigenvector of a 2 × 2 matrix
- 7367 · core · C10: An eigenvector of a 2 × 2 matrix
- 9FM0 · FP2 · 3.1 Further matrix algebra: Eigenvalues of a non-triangular matrix
- H245 · additional-pure · Matrices: Eigenvalues of a non-triangular matrix
- H645 · Y435 · Matrices: Eigenvalues of a non-triangular matrix
- 7367 · core · C10: Eigenvalues of a non-triangular matrix
- 9FM0 · FP2 · 3.1 Further matrix algebra: An eigenvector of a non-triangular matrix
- H245 · additional-pure · Matrices: An eigenvector of a non-triangular matrix
- H645 · Y435 · Matrices: An eigenvector of a non-triangular matrix
- 7367 · core · C10: An eigenvector of a non-triangular matrix
- 9FM0 · FP2 · 3.1 Further matrix algebra: A matrix power acting on a vector
- H245 · additional-pure · Matrices: A matrix power acting on a vector
- H645 · Y435 · Matrices: A matrix power acting on a vector
- 7367 · core · C10: A matrix power acting on a vector
Next practice: Inverse matrices.