Diagonalisation of Matrices: Explained with Examples

Diagonalisation writes a square matrix as A = P D P inverse with D diagonal and P invertible. Then the transformation is transparent: P changes to an eigenvector basis, D scales each coordinate by an eigenvalue, P inverse changes back. Powers and long-term behaviour, hard for A directly, become trivial for D.

This matters because many problems iterate a linear map: population models, Markov chains, oscillators and ranking algorithms all ask what A to the kth does for large k. Along eigenvectors, A to the kth just raises eigenvalues to the kth power, turning dynamics into diagonal arithmetic.

Key Concepts

• Diagonalisable: A = P D P inverse for some invertible P and diagonal D.
• Criterion: n independent eigenvectors, equivalently minimal polynomial with distinct linear factors, equivalently geometric = algebraic multiplicity everywhere.
• Construction: columns of P are the eigenvectors; diagonal of D holds matching eigenvalues in the same order.
• Defective: too few eigenvectors, e.g. [[2,1],[0,2]], never diagonalisable.
• Sufficient condition: n distinct eigenvalues always work, since their eigenvectors are automatically independent.

Formulas

• Factorisation: A = P D P inverse with D = diag(lambda 1, …, lambda n).
• Powers: A to the kth = P diag(lambda 1 to the kth, …, lambda n to the kth) P inverse.
• Check: verify A P = P D instead of computing P inverse; column j of each side is lambda j times column j of P.

Worked Example

Diagonalise A = [[4,1],[2,3]]: eigenvalues 5, 2 with eigenvectors (1,1), (1,minus 2). Set P = [[1,1],[1,minus 2]], D = diag(5,2). Verify AP = PD: AP column 1 = (4,2)+(1,3) = (5,5); column 2 = (4,2) minus 2(1,3) = (2,minus 4). PD scales P's columns by 5 and 2: [[5,2],[5,minus 4]]. They match, so A = PDP inverse. Bonus: A cubed = P diag(125,8) P inverse, since 5 cubed = 125 and 2 cubed = 8.

Common Misconceptions

• P's columns can be in any order. Column j of P must match the eigenvalue in D's (j,j) position, or AP = PD fails.
• Symmetric matrices may fail. Real symmetric matrices always diagonalise with orthonormal P (spectral theorem).
• It works over any field. The characteristic polynomial must split over your field; rotations diagonalise over C, not R.

Real-World Applications

• Markov chains: long-run behaviour comes from the dominant eigenvalue in D.
• Vibrations: normal modes are eigenvectors; diagonalisation decouples the equations.
• PCA: diagonalising the covariance matrix finds uncorrelated directions of greatest variance.

Diagonalisation factors a matrix into basis-change, pure scaling, basis-change-back. Stack n independent eigenvectors as P's columns with eigenvalues on D in matching order, and confirm with AP = PD.