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.
- Say what repeated and complex eigenvalues mean for the transformation.
COMMON MISCONCEPTION
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: most transformations leave some directions unturned. Rearranging to (M − λI)v = 0 needs a singular matrix if v is to be non-zero, and that 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, built from the trace 7 and the 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
For a 3 × 3 matrix the characteristic equation is a cubic, so there are three eigenvalues counted with repeats. Substitute each back into (M − λI)v = 0 and the system always has a free parameter, since the matrix is singular by construction. Any non-zero solution will do, because 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. What remains is the 2 × 2 block with rows (3, 4) and (4, 9), of trace 12 and determinant 11, so λ² − 12λ + 11 = 0 and λ = 1 or 11.
Write the block's variables as y and z. For λ = 11: −8y + 4z = 0, giving (0, 1, 2). For λ = 1: 2y + 4z = 0, giving (0, 2, −1).
The eigenvalues are 1, 2 and 11. Notice (0, 1, 2)·(0, 2, −1) = 0. A symmetric matrix always has perpendicular eigenvectors when the eigenvalues are different, as all three are here, and that gives you a free check.
Normalising, repeats and complex pairs
Normalising divides an eigenvector by its length to give a unit vector, so (0, 1, 2) becomes (0, 1/√5, 2/√5). Questions ask for this when the eigenvectors are about to be used as the columns of an orthogonal matrix, and the surds are unavoidable.
A repeated eigenvalue can come with a whole plane of eigenvectors instead of a single line, and then you are free to pick any two independent directions from that plane. Complex eigenvalues signal a rotation, where no real direction survives unturned at all, and a 2 × 2 rotation matrix is the standard example.
GUIDED PRACTICE
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 gives (λ − 1)(λ − 6) = 0, so λ = 1 or 6.
The sum of the eigenvalues is the trace and their product is the determinant, which checks both roots in one line.
ASSESSMENT FOCUS
- Write det(M − λI) = 0 explicitly before expanding; that line carries the method mark.
- For a 2 × 2, quote λ² − (trace)λ + determinant = 0 and save yourself the expansion.
- Any non-zero multiple of an eigenvector is an eigenvector, so say so instead of hunting for a canonical one.
- Check every pair by computing Mv and comparing with λv. It takes seconds and catches sign slips.
- Normalise only when the question asks, and leave the surds in exact form when you do.
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
Trace = 3 + (−4) = −1 and determinant = 3 × (−4) = −12. The eigenvalues fix both, even without seeing the matrix.
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 whenever the eigenvalues differ.
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.
Or read them with their worked answers on the eigenvalues and eigenvectors questions page.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- 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.
- Say what repeated and complex eigenvalues mean for the transformation.
Open the full revision checklist to see every objective in the course in one place.