Maths › Further Pure 2 › Eigenvalues and eigenvectors
Eigenvalues and eigenvectors
Most vectors are turned by a matrix. A few are merely stretched, and those few reveal what the transformation is really doing underneath the coordinates.
Builds on Determinants and inverses and Systems of equations and invariance.
IN THIS TOPIC
- Form and solve the characteristic equation of a 2 × 2 or 3 × 3 matrix.
- Find eigenvectors for each eigenvalue, and normalise them when asked.
- Interpret eigenvectors as the directions a transformation leaves alone.
WHAT YOU PROBABLY THINK
Every vector is turned by a matrix transformation, since that is what transformations do.
Directions that survive
An eigenvector is a non-zero vector v with Mv = λv: the transformation stretches it by the scalar eigenvalue λ and moves it nowhere else. Such directions almost always exist, so the opening claim overstates the case. Rearranging to (M − λI)v = 0 needs a singular matrix for a non-zero solution, which gives the characteristic equation:
WORKED EXAMPLE
Both eigen-pairs of a 2 × 2
Find the eigenvalues and eigenvectors of the matrix with rows (4, 1) and (2, 3).
The characteristic equation is λ² − 7λ + 10 = 0, using the trace 7 and determinant 10, so λ = 5 or λ = 2.
For λ = 5: (4 − 5)x + y = 0 gives y = x, so v = (1, 1) and indeed Mv = (5, 5).
For λ = 2: 2x + y = 0 gives y = −2x, so v = (1, −2) and Mv = (2, −4) = 2v.
These are the invariant lines from the core matrices unit, now with their stretch factors named.
Three dimensions, and tidying up
For a 3 × 3 matrix the characteristic equation is a cubic, so there are three eigenvalues counted with repeats. Substituting each back into (M − λI)v = 0 leaves a system with a free parameter, since the matrix is singular by construction: any non-zero solution will do, and an eigenvector is only ever determined up to a scalar multiple.
WORKED EXAMPLE
A symmetric 3 × 3
Find the eigenvalues and eigenvectors of the matrix with rows (2, 0, 0), (0, 3, 4) and (0, 4, 9).
The first row and column isolate the direction (1, 0, 0) with λ = 2. The rest reduces to the 2 × 2 block with rows (3, 4) and (4, 9): trace 12, determinant 11, so λ² − 12λ + 11 = 0 and λ = 1 or 11.
For λ = 11: −8x + 4y = 0 in the block, giving (0, 1, 2). For λ = 1: 2x + 4y = 0, giving (0, 2, −1).
The eigenvalues are 1, 2 and 11. Note (0, 1, 2)·(0, 2, −1) = 0: a symmetric matrix always has perpendicular eigenvectors, which is a free check.
Normalising divides an eigenvector by its length, giving a unit vector: (0, 1, 2) becomes (0, 1/√5, 2/√5). Repeated eigenvalues may come with a whole plane of eigenvectors rather than a single line, and complex eigenvalues signal a rotation, with no real direction left unturned at all.
YOUR TURN
A quick characteristic equation
Find the eigenvalues of the matrix with rows (5, 2) and (2, 2).
Show the working
Trace 7 and determinant 10 − 4 = 6, so λ² − 7λ + 6 = 0.
Factorising: (λ − 1)(λ − 6) = 0, so λ = 1 or 6.
The sum of eigenvalues is the trace and their product is the determinant, which checks both roots in one line.
THE EXAM BIT
- Write det(M − λI) = 0 explicitly before expanding; that line carries the method mark.
- For a 2 × 2, the characteristic equation is λ² − (trace)λ + determinant = 0: quote it and save time.
- Any non-zero multiple of an eigenvector is an eigenvector; say so rather than hunting for a canonical one.
- Check every pair by computing Mv and comparing with λv; it takes seconds and catches sign slips.
CHECK YOURSELF
A 2 × 2 matrix has eigenvalues 3 and −4. Write down its trace and determinant.
Show a hint
The characteristic equation is λ² − (trace)λ + det = 0.
Show the answer
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Eigenvectors satisfy Mv = λv; solve det(M − λI) = 0 for the eigenvalues, then substitute back.
Trace is the sum of the eigenvalues and determinant their product; symmetric matrices have perpendicular eigenvectors.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Form and solve the characteristic equation of a 2 × 2 or 3 × 3 matrix.
- Find eigenvectors for each eigenvalue, and normalise them when asked.
- Interpret eigenvectors as the directions a transformation leaves alone.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.