Editing Definite matrix (section)

== Other characterizations ==
Let <math>M</math> be an <math>n \times n</math> [[Hermitian matrix|real symmetric matrix]], and let <math>B_1(M) \equiv \{ \mathbf{x} \in \mathbb{R}^n : \mathbf{x}^\mathsf{T} M\mathbf{x} \leq 1\}</math> be the "unit ball" defined by <math>M.</math> Then we have the following
* <math>B_1( \mathbf{v}\mathbf{v}^\mathsf{T} )</math> is a solid slab sandwiched between <math>\pm \{ \mathbf{w}: \langle \mathbf{w}, \mathbf{v}\rangle = 1 \}.</math>
* <math>M \succeq 0</math> if and only if <math>B_1(M)</math> is an ellipsoid, or an ellipsoidal cylinder.
* <math>M \succ 0</math> if and only if <math>B_1(M)</math> is bounded, that is, it is an ellipsoid.
* If <math>N \succ 0,</math> then <math>M \succeq N</math> if and only if <math>B_1(M) \subseteq B_1(N);</math> <math>M \succ N</math> if and only if <math>B_1(M) \subseteq \operatorname{int}\bigl(B_1(N)\bigr).</math>
* If <math>N \succ 0,</math> then <math>M \succeq \frac{ \mathbf{v}\mathbf{v}^\mathsf{T} }{\mathbf{v}^\mathsf{T} N\mathbf{v}}</math> for all <math>v \neq 0</math> if and only if <math display="inline">B_1(M) \subset \bigcap_{ \mathbf{v}^\mathsf{T} N\mathbf{v} = 1 } B_1(\mathbf{v} \mathbf{v}^\mathsf{T}).</math> So, since the polar dual of an ellipsoid is also an ellipsoid with the same principal axes, with inverse lengths, we have <math display="block">B_1(N^{-1}) = \bigcap_{\mathbf{v}^\mathsf{T} N\mathbf{v} = 1} B_1(\mathbf{v}\mathbf{v}^\mathsf{T}) = \bigcap_{ \mathbf{v}^\mathsf{T} N\mathbf{v} = 1 } \{ \mathbf{w}: |\langle \mathbf{w}, \mathbf{v}\rangle| \leq 1 \}.</math> That is, if <math>N</math> is positive-definite, then <math>M \succeq \frac{ \mathbf{v} \mathbf{v}^\mathsf{T} }{\mathbf{v}^\mathsf{T} N\mathbf{v}}</math> for all <math>\mathbf{v} \neq \mathbf{0}</math> if and only if <math>M \succeq N^{-1} .</math>

Let <math>M</math> be an <math>n \times n</math> [[Hermitian matrix]]. The following properties are equivalent to <math>M</math> being positive definite:
; The associated sesquilinear form is an inner product : The [[sesquilinear form]] defined by <math>M</math> is the function <math>\langle \cdot, \cdot \rangle</math> from <math>\mathbb{C}^n \times \mathbb{C}^n</math> to <math>\mathbb{C}^n</math> such that <math>\langle \mathbf{x}, \mathbf{y} \rangle \equiv \mathbf{y}^* M\mathbf{x}</math> for all <math>\mathbf{x}</math> and <math>\mathbf{y}</math> in <math>\mathbb{C}^n,</math> where <math>\mathbf{y}^*</math> is the conjugate transpose of <math>\mathbf{y}.</math> For any complex matrix <math>M,</math> this form is linear in <math>x</math> and semilinear in <math>\mathbf{y}.</math> Therefore, the form is an [[inner product]] on <math>\mathbb{C}^n</math> if and only if <math>\langle \mathbf{z}, \mathbf{z} \rangle</math> is real and positive for all nonzero <math>\mathbf{z};</math> that is if and only if <math>M</math> is positive definite. (In fact, every inner product on <math>\mathbb{C}^n</math> arises in this fashion from a Hermitian positive definite matrix.)
; Its leading principal minors are all positive : The {{mvar|k}}th [[minor (linear algebra)|leading principal minor]] of a matrix <math>M</math> is the [[determinant]] of its upper-left <math>k \times k</math> sub-matrix. It turns out that a matrix is positive definite if and only if all these determinants are positive. This condition is known as [[Sylvester's criterion]], and provides an efficient test of positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an [[upper triangular matrix]] by using [[elementary row operations]], as in the first part of the [[Gaussian elimination]] method, taking care to preserve the sign of its determinant during [[pivot element|pivoting]] process.  Since the {{mvar|k}}th leading principal minor of a triangular matrix is the product of its diagonal elements up to row <math>k,</math> Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.  This condition can be checked each time a new row <math>k</math> of the triangular matrix is obtained.

A positive semidefinite matrix is positive definite if and only if it is [[invertible matrix|invertible]].<ref>{{harvtxt|Horn|Johnson|2013}}, p. 431, Corollary 7.1.7</ref>
A matrix <math>M</math> is negative (semi)definite if and only if <math>-M</math> is positive (semi)definite.