Why Covariance Matrix of Complex Random Vector is Hermitian Positive Definite

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Discussion Overview

The discussion revolves around the properties of the covariance matrix of complex random vectors, specifically questioning why it is considered Hermitian positive definite rather than merely positive semi-definite. Participants explore definitions, implications for Gaussian and non-Gaussian random variables, and the characteristics of crosscovariance matrices.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant expresses difficulty in finding a proof that the covariance matrix of a complex random vector is Hermitian positive definite, questioning the triviality of such a proof.
  • Another participant notes that every real symmetric matrix is Hermitian and mentions a complex to real isomorphism for Gaussian random variables, but is uncertain if this applies to non-Gaussian random variables.
  • There is a reference to a source that states the covariance matrix is positive semi-definite, while another source claims it is always positive definite, highlighting a potential contradiction.
  • A participant questions whether the property of being Hermitian positive definite also applies to crosscovariance matrices, suggesting that there are no constraints on the elements of a crosscovariance matrix, which may affect its definiteness.

Areas of Agreement / Disagreement

Participants express differing views on the definiteness of covariance matrices, with some asserting positive definiteness and others suggesting positive semi-definiteness. The applicability of these properties to crosscovariance matrices remains contested.

Contextual Notes

There are unresolved questions regarding the definitions and properties of covariance and crosscovariance matrices, particularly in relation to Gaussian versus non-Gaussian random variables. The discussion reflects uncertainty about the implications of these properties.

fcastillo
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I've been reading everywhere, including wikipedia, and I can't seem to find a prove to the fact that the covariance matrix of a complex random vector is Hermitian positive definitive. Why is it definitive and not just simple semi-definitive like any other covariance matrix?
Wikipedia just states this and never provides with a prove. Some people might say that it follows from the definition and that a prove is trivial, but I just can't seem to find why. Even if the prove is trivial (and it's eluding me) can somebody please demonstrate why?

Also, if this is true, does it also apply for crosscovariance matrix? (between two different complex random vectors)
 
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Every real symmetric matrix is Hermitian and there is a complex to real isomorphism for Gaussian RVs. I believe this arises from the definition of Hermitian matrices. I'm not sure if the isomorphism holds for the covariance matrix of non-Gaussian RVs.

http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/expect.html
 
Last edited:
SW VandeCarr said:
Every real symmetric matrix is Hermitian and there is a complex to real isomorphism for Gaussian RVs. I believe this arises from the definition of Hermitian matrices. I'm not sure if the isomorphism holds for the covariance matrix of non-Gaussian RVs.

http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/expect.html

Thanks for the reply, in that post it states that the matrix is positive semi-definite. And the reference I found somewhere else said that the covariance matrix is always positive-definite. Thanks a lot for your reply!
 
fcastillo said:
Thanks for the reply, in that post it states that the matrix is positive semi-definite. And the reference I found somewhere else said that the covariance matrix is always positive-definite. Thanks a lot for your reply!

Sorry, for some reason the wrong link came up. I've got one that answers your question re covariance matrices.

http://www.riskglossary.com/link/positive_definite_matrix.htm
 
Last edited by a moderator:
fcastillo said:
Also, if this is true, does it also apply for crosscovariance matrix? (between two different complex random vectors)

There are no constraints on the elements of a crosscovariance matrix, so it is clearly not necessarily positive semidefinite (or even square in shape!)
 

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