Editing Definite matrix (section)

== Simultaneous diagonalization ==
One symmetric matrix and another matrix that is both symmetric and positive definite can be [[diagonalizable matrix#Simultaneous diagonalization|simultaneously diagonalized]]. This is so  although simultaneous diagonalization is not necessarily performed with a [[Matrix similarity|similarity transformation]]. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.

Let <math>M</math> be a symmetric and <math>N</math> a symmetric and positive definite matrix. Write the generalized eigenvalue equation as <math>\left(M - \lambda N\right)\mathbf{x} = 0</math> where we impose that <math>\mathbf{x}</math> be normalized, i.e. <math>\mathbf{x}^\mathsf{T} N \mathbf{x} = 1.</math> Now we use [[Cholesky decomposition]] to write the inverse of <math>N</math> as <math>Q^\mathsf{T} Q.</math> Multiplying by <math>Q</math> and letting <math>\mathbf{x} = Q^\mathsf{T} \mathbf{y},</math> we get <math>Q \left(M - \lambda N\right) Q^\mathsf{T} \mathbf{y} = 0,</math> which can be rewritten as <math>\left(Q M Q^\mathsf{T} \right)\mathbf{y} = \lambda \mathbf{y}</math> where <math>\mathbf{y}^\mathsf{T} \mathbf{y} = 1.</math> Manipulation now yields <math>MX = NX\Lambda</math> where <math>X</math> is a matrix having as columns the generalized eigenvectors and <math>\Lambda</math> is a diagonal matrix of the generalized eigenvalues. Now premultiplication with <math>X^\mathsf{T}</math> gives the final result: <math>X^\mathsf{T} MX = \Lambda</math> and <math>X^\mathsf{T} N X = I,</math> but note that this is no longer an orthogonal diagonalization with respect to the inner product where <math>\mathbf{y}^\mathsf{T} \mathbf{y} = 1.</math> In fact, we diagonalized <math>M</math> with respect to the inner product induced by <math>N.</math><ref>{{harvtxt|Horn|Johnson|2013}}, p. 485, Theorem 7.6.1</ref>

Note that this result does not contradict what is said on simultaneous diagonalization in the article [[diagonalizable matrix#Simultaneous diagonalization|Diagonalizable matrix]], which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other.