Editing Definite matrix (section)

== Eigenvalues ==
Let <math>M</math> be an <math>n \times n</math> [[Hermitian matrix]] (this includes real [[symmetric matrices]]). All eigenvalues of <math>M</math> are real, and their sign characterize its definiteness:
* <math>M</math> is positive definite if and only if all of its eigenvalues are positive.
* <math>M</math> is positive semi-definite if and only if all of its eigenvalues are non-negative.
* <math>M</math> is negative definite if and only if all of its eigenvalues are negative.
* <math>M</math> is negative semi-definite if and only if all of its eigenvalues are non-positive.
* <math>M</math> is indefinite if and only if it has both positive and negative eigenvalues.

Let <math>P D P^{-1}</math> be an [[eigendecomposition of a matrix|eigendecomposition]] of <math>M,</math> where <math>P</math> is a [[unitary matrix|unitary complex matrix]] whose columns comprise an [[orthonormal basis]] of [[eigenvector]]s of <math>M,</math> and <math>D</math> is a ''real'' [[diagonal matrix]] whose [[main diagonal]] contains the corresponding [[eigenvalue]]s. The matrix <math>M</math> may be regarded as a diagonal matrix <math>D</math> that has been re-expressed in coordinates of the (eigenvectors) basis <math>P.</math> Put differently, applying <math>M</math> to some vector <math>\mathbf{z},</math> giving <math>M \mathbf{z},</math> is the same as [[Change of basis|changing the basis]] to the eigenvector coordinate system using <math>P^{-1},</math> giving <math>P^{-1} \mathbf{z},</math> applying the [[Transformation matrix|stretching transformation]] <math>D</math> to the result, giving <math>D P^{-1} \mathbf{z},</math> and then changing the basis back using <math>P,</math> giving <math>P D P^{-1} \mathbf{z}.</math>

With this in mind, the one-to-one change of variable <math>\mathbf{y} = P\mathbf{z}</math> shows that <math>\mathbf{z}^* M\mathbf{z}</math> is real and positive for any complex vector <math>\mathbf{z}</math> if and only if <math>\mathbf{y}^* D \mathbf{y}</math> is real and positive for any <math>y;</math> in other words, if <math>D</math> is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal – that is, every eigenvalue of <math>M</math> – is positive. Since the [[spectral theorem]] guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using [[Descartes' rule of signs|Descartes' rule of alternating signs]] when the [[characteristic polynomial]] of a real, symmetric matrix <math>M</math> is available.