diagonalised

In linear algebra, a matrix is said to be diagonalised if it can be expressed as the product of an invertible matrix, a diagonal matrix, and the inverse of the invertible matrix. Mathematically, a square matrix A is diagonalised if there exists an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹. The diagonal matrix D contains the eigenvalues of A on its main diagonal, and the columns of the matrix P are the corresponding eigenvectors of A.

The process of finding P and D is called diagonalisation. Not all matrices can be diagonalised. A necessary and sufficient condition for a matrix to be diagonalisable is that it must have a full set of linearly independent eigenvectors. This is guaranteed for symmetric matrices, and also for matrices with distinct eigenvalues. If a matrix is diagonalisable, its diagonal form D is unique up to the order of the diagonal entries.

Diagonalisation is a powerful tool in linear algebra. It simplifies many matrix operations. For instance, computing powers of a diagonalisable matrix becomes much easier: Aᵏ = PDᵏP⁻¹, and Dᵏ is simply the diagonal matrix with the k-th powers of the original eigenvalues on the diagonal. This property is crucial in solving systems of linear differential equations, analysing Markov chains, and in various areas of physics and engineering. The concept also extends to linear transformations, where a linear transformation is diagonalisable if there exists a basis in which its matrix representation is diagonal.