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== 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}^\top 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}^\top )</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}^\top }{\ \mathbf{v}^\top 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}^\top N\ \mathbf{v} = 1 } B_1(\mathbf{v} \mathbf{v}^\top) ~.</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}^\top N\ \mathbf{v} = 1} B_1(
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.
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