Definite matrix: Difference between revisions

Revision as of 23:07, 29 January 2025

Revision as of 09:27, 7 February 2025

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|title = Parameter Estimation for Scientists and Engineers |edition=online

|publisher = John Wiley & Sons

|ISBNisbn = 978-047-017386-2

|pages = 259–263

|doi = 10.1002/9780470173862 |doi-access=

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<math display="block"> \max_{i,j} \left|m_{ij}\right| \leq \max_i m_{ii}</math>

An <math>n \times n</math> Hermitian matrix <math>M</math> is positive definite if it satisfies the following trace inequalities:<ref>{{cite journal | title=Bounds for Eigenvalues using Traces | lastlast1=Wolkowicz | firstfirst1=Henry | last2 = Styan | first2 = George P.H. | journal=Linear Algebra and itsIts Applications | issue=29 | publisher=Elsevier | year=1980 | volume=29 | pages=471–506 | doi=10.1016/0024-3795(80)90258-X }}</ref>

<math display="block">\operatorname{tr}(M) > 0 \quad \mathrm{and} \quad \frac{(\operatorname{tr}(M))^2}{\operatorname{tr}(M^2)} > n-1 ~.</math>

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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>

===Kronecker product===

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* {{cite journal

|lastlast1=Bernstein |firstfirst1=B.

|last2=Toupin |first2=R.A.

|year=1962