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Eigenvalues and eigenvectors

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

An eigenvector of a matrix is a non-zero vector whose direction is unchanged by the transformation: it is only stretched, by a factor called the eigenvalue. Eigenvectors give the invariant lines through the origin.

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Definition and characteristic equation

Ax = λx with x ≠ 0. Then (A - λI)x = 0 has a non-zero solution, so A - λI must be singular:

det(A - λI) = 0, the characteristic equation.

For a 2 × 2 matrix this is a quadratic, λ2 - (trace)λ + det = 0. For a 3 × 3 matrix it is a cubic; look for a factor by trying small integers or expand along a row or column with zeros.

Finding eigenvectors

For each eigenvalue λ, solve (A - λI)x = 0. The equations are dependent, so you get a whole line of solutions: choose a simple non-zero vector.

For a 3 × 3 matrix, a neat trick is that the vector product of two rows of A - λI is an eigenvector (if the rows are not parallel).

Normalised eigenvectors and checks

A normalised eigenvector has length 1: divide by its magnitude, e.g. (1, 1) becomes (1/sqrt(2), 1/sqrt(2)).

Checks: the eigenvalues add up to the trace and multiply to the determinant. Each eigenvector x gives an invariant line r = tx through O.

Worked example

Find the eigenvalues and eigenvectors of A with rows (4, 1), (2, 3).

  1. det(A - λI) = (4 - λ)(3 - λ) - 2 = λ2 - 7λ + 10 = 0, so λ = 2 or 5
  2. λ = 5: -x + y = 0, so x = (1, 1)
  3. λ = 2: 2x + y = 0, so x = (1, -2)

Answer: λ = 5 with (1, 1); λ = 2 with (1, -2)

Key idea: Solve det(A - λI) = 0 for λ, then (A - λI)x = 0 for x. Eigenvalues sum to the trace and multiply to the determinant. Normalise by dividing by the length.

Check yourself

  1. Find the eigenvalues of the matrix with rows (4, 1), (2, 3).
    Show answer

    2, 5. λ2 - 7λ + 10 = 0 gives (λ - 2)(λ - 5) = 0.

  2. Which is an eigenvector of the matrix with rows (4, 1), (2, 3) for the eigenvalue 5?
    • (1, -2)
    • (1, 1)
    • (2, 1)
    • (1, 5)
    Show answer

    (1, 1). (4 + 1, 2 + 3) = (5, 5) = 5(1, 1).

  3. A has rows (3, 0, 1), (2, 2, 1), (0, 0, 1). Given that (1, 2, 0) is an eigenvector, find its eigenvalue.
    Show answer

    3. A(1, 2, 0) = (3, 2 + 4, 0) = (3, 6, 0) = 3(1, 2, 0).

  4. Find all three eigenvalues of A with rows (3, 0, 1), (2, 2, 1), (0, 0, 1).
    Show answer

    1, 2, 3. Expand det(A - λI) along the bottom row: (1 - λ)[(3 - λ)(2 - λ) - 0] = 0, so λ = 1, 2 or 3.

  5. A 3 × 3 matrix has trace 9 and two of its eigenvalues are 2 and 3. Find the third eigenvalue.
    Show answer

    4. The eigenvalues add up to the trace: 9 - 2 - 3 = 4.

  6. Normalise the eigenvector (1, 1). Give its first component to 3 decimal places.
    Show answer

    0.707. Length sqrt(2), so the unit vector is (1/sqrt(2), 1/sqrt(2)) and 1/sqrt(2) = 0.707.

  7. What does an eigenvector x of a transformation matrix A represent geometrically?
    • A point that is fixed by the transformation
    • The axis of every rotation
    • A line through the origin that is mapped onto itself
    • A vector that is mapped to zero
    Show answer

    A line through the origin that is mapped onto itself. Ax = λx, so every multiple of x is mapped to a multiple of x: the line r = tx is invariant.

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Last updated 8 November 2026. Spotted a mistake? Tell us.