Diagonalcapable

Diagonalcapable is a term used in linear algebra to describe a square matrix that is similar to a diagonal matrix. Two matrices, A and B, are said to be similar if there exists an invertible matrix P such that B = P⁻¹AP. A matrix is diagonal if all of its off-diagonal elements are zero. A diagonal matrix has its eigenvalues on the main diagonal and zeros elsewhere.

A matrix is diagonalizable if and only if there exists a basis of eigenvectors for the vector space. This means that the number of linearly independent eigenvectors for the matrix must be equal to its dimension. Equivalently, for each eigenvalue, the algebraic multiplicity (the number of times the eigenvalue is a root of the characteristic polynomial) must equal its geometric multiplicity (the dimension of the eigenspace corresponding to that eigenvalue).

If a matrix A is diagonalisable, then it can be written as A = PDP⁻¹, where D is a diagonal matrix whose entries are the eigenvalues of A, and P is an invertible matrix whose columns are the corresponding linearly independent eigenvectors of A. The process of finding P and D is called diagonalization. Diagonalizable matrices have many useful properties, particularly in solving systems of linear differential equations and in simplifying matrix powers. Not all square matrices are diagonalisable.