Editing Diagonalizable matrix (section)

== Simultaneous diagonalization ==
{{See also|Triangular matrix#Simultaneous triangularisability|l1=Simultaneous triangularisability|Weight (representation theory)|Positive definite matrix#Simultaneous_diagonalization|l3=Positive definite matrix}}

A set of matrices is said to be ''simultaneously diagonalizable'' if there exists a single invertible matrix <math>P</math> such that <math>P^{-1}AP</math> is a diagonal matrix for every <math>A</math> in the set. The following theorem characterizes simultaneously diagonalizable matrices: A set of diagonalizable [[Commuting matrices|matrices commutes]] if and only if the set is simultaneously diagonalizable.<ref name="HornJohnson">{{cite book|title=Matrix Analysis, second edition|last1=Horn|first1=Roger A.|last2=Johnson|first2=Charles R.|publisher=Cambridge University Press|year=2013|isbn=9780521839402}}</ref>{{rp|p. 64}}

The set of all <math>n \times n</math> diagonalizable matrices (over {{nowrap|<math>\Complex</math>)}} with <math>n > 1</math> is not simultaneously diagonalizable. For instance, the matrices

:<math> \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} </math>

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

A set consists of commuting [[normal matrix|normal matrices]] if and only if it is simultaneously diagonalizable by a [[unitary matrix]]; that is, there exists a unitary matrix <math>U</math> such that <math>U^{*} AU</math> is diagonal for every <math>A</math> in the set.

In the language of [[Lie theory]], a set of simultaneously diagonalizable matrices generates a [[toral Lie algebra]].