Diagonalisation and Powers of a Matrix
A matrix with a basis of eigenvectors becomes diagonal in that basis. Its diagonal form lets us compute powers by raising the eigenvalues.
What a learner can do afterwards
- Diagonalise a matrix and identify the eigenvector matrix P and diagonal eigenvalue matrix D
- Compute a high power of a matrix through its diagonal form
- Give a matrix that cannot be diagonalised and say what fails
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For real diagonalisation, write A = P D P inverse. P must be invertible, with a basis of real eigenvectors as its columns. D has their matching eigenvalues on its diagonal, in the same order. The relation A P = P D checks each column's eigenvector equation. A diagonal matrix already displays its eigenvalues: diag(5, 8) has eigenvalues 5 and 8.
For a positive integer k, A to the k = P D to the k P inverse. Each adjacent P inverse P cancels to the identity. Raise only D's diagonal entries to the kth power. Trace means the sum of diagonal entries and is unchanged by a change of basis. If D = diag(3, 4), trace(A squared) = 9 + 16 = 25.
An n by n matrix is diagonalisable over the reals exactly when it has n independent real eigenvectors. Having n distinct real eigenvalues is sufficient. Repeated eigenvalues can also work. For A = [[r, h], [0, r]] with h nonzero, the only eigenvalue is r. The equation (A minus rI)v = 0 forces v's second component to zero. Its eigenspace has dimension one, not two.
For A = [[3, 2], [0, 5]], A(1, 0) = 3(1, 0) and A(1, 1) = 5(1, 1). These independent columns give P = [[1, 1], [0, 1]], D = diag(3, 5), and P inverse = [[1, -1], [0, 1]]. Multiplication gives P D P inverse = A. Swapping P's columns requires swapping D's diagonal entries too.
For P = [[1, 1], [0, 1]] and D = diag(a, b), A to the k has diagonal entries a to the k and b to the k. Its top-right entry is b to the k minus a to the k. For any diagonalisable 2 by 2 A, trace(A to the k) is the sum of its eigenvalues raised to k, since changing basis preserves trace. Distinct real eigenvalues give real diagonalisation. In eigenvector coordinates, each application multiplies each coordinate by its eigenvalue.
An eigenvalue's algebraic multiplicity counts its occurrences as a root of the characteristic polynomial. Its eigenspace dimension counts independent eigenvectors for that eigenvalue and cannot exceed its multiplicity. Diagonalisation needs equality for every eigenvalue. In a 3 by 3 matrix, a double eigenvalue with eigenspace dimension one and a simple eigenvalue can supply at most two independent eigenvectors.
Put eigenvectors into P and eigenvalues into D, then use the diagonal form to compute powers.
Lesson and assessment text authored for this dataset.
Readings and additional questions authored for this dataset. Linked videos by Khan Academy provide eigenvalue and eigenvector refreshers. Video reuse terms are unknown.
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