Diagonalisability

Diagonalisability refers to the property of a square matrix or linear operator that can be transformed into a diagonal matrix through a similarity transformation. A matrix A is diagonalisable if there exists an invertible matrix P and a diagonal matrix D such that A = P D P⁻¹. In this context, the columns of P are the eigenvectors of A, and the diagonal entries of D are the corresponding eigenvalues.

For a matrix to be diagonalisable, it must satisfy two key conditions: it must have a full set of linearly independent eigenvectors, and the geometric multiplicity of each eigenvalue must equal its algebraic multiplicity. The algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace associated with that eigenvalue.

A matrix is guaranteed to be diagonalisable if it has n linearly independent eigenvectors in an n-dimensional space, where n is the size of the matrix. This is always true for matrices with distinct eigenvalues, as eigenvectors corresponding to distinct eigenvalues are linearly independent. However, even if eigenvalues are repeated, diagonalisability may still hold if the eigenspaces span the entire space.

Diagonalisation simplifies many computations, such as matrix exponentiation, polynomial evaluation, and solving systems of linear differential equations. The diagonal form D allows operations like raising A to a power or applying a function to A to be performed element-wise, which is computationally efficient.

Not all matrices are diagonalisable. For example, a Jordan block matrix with repeated eigenvalues and a single Jordan chain is not diagonalisable because it lacks a full set of linearly independent eigenvectors. In such cases, the matrix can instead be transformed into a Jordan normal form, which is a block-diagonal matrix with diagonal blocks corresponding to Jordan chains.