Maths (university) · Part 10
Eigenvalues and eigenvectors
The special directions a matrix only stretches, and why they matter.
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- Eigenvector and eigenvalueAn eigenvector of A is a non-zero vector v with A v = lambda v. The matrix only stretches it, and the stretch factor lambda is the eigenvalue.
- Characteristic equationThe eigenvalues of A are the solutions of det(A - lambda I) = 0. This gives a polynomial in lambda.
- Finding eigenvectorsFor each eigenvalue lambda, solve (A - lambda I) v = 0. Any non-zero solution is an eigenvector, and any scalar multiple of it is too.
- Trace and determinant checkThe eigenvalues add up to the trace (sum of diagonal entries) and multiply to the determinant. Use this to check your work.
- DiagonalisationIf A has enough independent eigenvectors, then A = P D P^(-1). The columns of P are the eigenvectors and D has the eigenvalues on its diagonal.
- Matrix powersWith A = P D P^(-1), the power A^n equals P D^n P^(-1). Raise each diagonal entry of D to the n-th power.
- Steady stateFor a Markov chain (columns of probabilities summing to 1), the long-run distribution is the eigenvector with eigenvalue 1.
- Symmetric matrixA matrix equal to its own transpose. Its eigenvalues are always real, and eigenvectors for different eigenvalues are perpendicular.
- Complex eigenvaluesSome real matrices have no real eigenvectors, because their characteristic equation has complex roots such as i and -i. Rotations do this.