Diagonalizable Matrix - Simultaneous Diagonalization

Simultaneous Diagonalization

See also: Simultaneous triangularisability and Weight (representation theory)

A set of matrices are said to be simultaneously diagonalisable if there exists a single invertible matrix P such that is a diagonal matrix for every A in the set. The following theorem characterises simultaneously diagonalisable matrices: A set of diagonalizable matrices commutes if and only if the set is simultaneously diagonalisable.

The set of all n-by-n diagonalisable matrices (over C) with n > 1 is not simultaneously diagonalisable. For instance, the matrices

are diagonalizable but not simultaneously diagonalizable because they do not commute.

A set consists of commuting normal matrices if and only if it is simultaneously diagonalisable by a unitary matrix; that is, there exists a unitary matrix U such that is diagonal for every A in the set.

In the language of Lie theory, a set of simultaneously diagonalisable matrices generate a toral Lie algebra.