Editing Definite matrix (section)

=== Hadamard product ===
If <math>M, N \geq 0,</math> although <math>M N</math> is not necessary positive semidefinite, the [[Hadamard product (matrices)|Hadamard product]] is, <math>M \circ N \geq 0</math> (this result is often called the [[Schur product theorem]]).<ref>{{harvtxt|Horn|Johnson|2013}}, p. 479, Theorem 7.5.3</ref>

Regarding the Hadamard product of two positive semidefinite matrices <math>M = (m_{ij}) \geq 0,</math> <math>N \geq 0,</math> there are two notable inequalities:
* Oppenheim's inequality: <math>\det(M \circ N) \geq \det (N) \prod\nolimits_i m_{ii}.</math><ref>{{harvtxt|Horn|Johnson|2013}}, p. 509, Theorem 7.8.16</ref>
* <math>\det(M \circ N) \geq \det(M) \det(N).</math><ref name=styan1973>{{cite journal |last=Styan |first=G.P. |year=1973 |title=Hadamard products and multivariate statistical analysis |journal=[[Linear Algebra and Its Applications]] |volume=6 |pages=217–240 |doi=10.1016/0024-3795(73)90023-2 }}, Corollary 3.6, p. 227</ref>