nondiagonalisable

In linear algebra, a square matrix is called nondiagonalisable if it cannot be expressed as the product of an invertible matrix and a diagonal matrix, and the inverse of the invertible matrix. More formally, a matrix A is nondiagonalisable if there is no invertible matrix P such that P⁻¹AP is a diagonal matrix. This property is deeply connected to the eigenvectors and eigenvalues of the matrix.

A key condition for diagonalisability is that the matrix must have a complete set of linearly independent eigenvectors. If the number of linearly independent eigenvectors is less than the dimension of the matrix, then the matrix is nondiagonalisable. This can occur when an eigenvalue has an algebraic multiplicity greater than its geometric multiplicity. The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the corresponding eigenspace (the set of all eigenvectors associated with that eigenvalue, plus the zero vector).

Nondiagonalisable matrices are also known as defective matrices. The Jordan normal form provides a way to represent any square matrix, including nondiagonalisable ones, in a nearly diagonal form. For a nondiagonalisable matrix, its Jordan normal form will contain at least one Jordan block of size greater than 1x1, which deviates from a purely diagonal structure. Understanding whether a matrix is diagonalisable is important in various applications, such as solving systems of differential equations and in the analysis of dynamical systems, as diagonalisable matrices simplify many computations.