Definite form

Description

A Bilineare Abbildung or Quadratic Form is called Positive Definite if inserting one vector only returns positive numbers. A matrix can be interpreted as a bilinear form, inducing a definition for definitiveness. Geometrically a matrix is called positive definite, when $x$ und $Ax$ have an acute angle.

Definition (Matrix)

Eine Matrix heißt

• positiv definit, wenn für alle $x\in {\mathbb{R}}^n$, $x^{\ast }Ax>0$ gilt.
• positiv semidefinit, wenn für alle $x\in {\mathbb{R}}^n$, $x^{\ast }Ax\ge 0$ gilt.
• negativ definit, wenn für alle $x\in {\mathbb{R}}^n$, $x^{\ast }Ax<0$ gilt.
• negativ semidefinit, wenn für alle $x\in {\mathbb{R}}^n$, $x^{\ast }Ax\le 0$ gilt.
• indefinit, wenn für $x\in {\mathbb{R}}^n$, sowohl positive als auch negative Werte angenommen werden.

Characterisation for matrices

A matrix $A$ is called

• positive definite when all eigenvalues are strictly positive
• positive semidefinite when all eigenvalues are nonnegative
• negative definite when all eigenvalues are strictly negative
• negative semidefinite when all eigenvalues are nonpositive
• indefinite when there are positive and negative eigenvalues

Properties