Diagonalisation

Diagonalisation is the process of expressing a square matrix A as a similar diagonal matrix. If there exists an invertible matrix P such that P^{-1} A P is diagonal, then A is diagonalizable and A can be written as A = P D P^{-1}, where D is diagonal. The diagonal entries of D are the eigenvalues of A, and the columns of P are the corresponding eigenvectors.

A matrix is diagonalizable over a field F if and only if it has n linearly independent eigenvectors in F^n (for an n × n matrix). Equivalently, the sum of the geometric multiplicities of its eigenvalues equals n, or the minimal polynomial of A splits into distinct linear factors over F. Over the complex field, the eigenvalues always exist; the requirement is having a full set of independent eigenvectors. Over the real field, A is diagonalizable over R if it has n real eigenvectors; otherwise a real diagonalisation is impossible and complex eigenvalues would appear in the diagonal only if complex entries are allowed.

Computationally, one finds the eigenvalues by solving det(A − λI) = 0, then for each eigenvalue λ_i solves (A − λ_i I) v = 0 to obtain eigenvectors v_i. If a full set of n independent eigenvectors is obtained, placing them as columns of P yields D = P^{-1} A P with D having λ_i on its diagonal.

Special cases include real symmetric matrices, which are orthogonally diagonalizable: there exists an orthogonal Q with Q^T A Q = D. If a matrix is not diagonalizable, it can be brought to Jordan normal form instead.

Applications of diagonalisation include simplifying powers A^k, solving linear differential equations and recurrence relations, and analysing long-term behavior of linear dynamical systems.