Why Covariance Matrix of Complex Random Vector is Hermitian Positive Definite

In summary, the covariance matrix of a complex random vector is Hermitian positive definite, as proven by the definition of Hermitian matrices and the isomorphism for Gaussian random variables. However, it is unclear if this also applies to non-Gaussian random variables. The crosscovariance matrix, on the other hand, does not have any constraints and can be of any shape, making it not necessarily positive semidefinite.
  • #1
fcastillo
2
0
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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  • #2
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
 
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  • #3
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!
 
  • #4
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
 
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  • #5
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!)
 

1. What is a covariance matrix?

A covariance matrix is a square matrix that summarizes the relationship between multiple variables in a dataset. It contains the variances of each variable along the diagonal and the covariances between each pair of variables in the off-diagonal elements.

2. How is a covariance matrix related to a complex random vector?

A complex random vector is a random variable with complex-valued elements. The elements of the covariance matrix represent the covariances between the real and imaginary parts of the elements in the complex random vector.

3. Why is the covariance matrix of a complex random vector Hermitian?

The covariance matrix of a complex random vector is Hermitian because it satisfies the property of conjugate symmetry. This means that the elements in the matrix are equal to their complex conjugates, which is necessary for the matrix to be self-adjoint.

4. What does it mean for a covariance matrix to be positive definite?

A positive definite covariance matrix has all positive eigenvalues, which means it is a real, symmetric matrix with no negative eigenvalues. This indicates that the data in the complex random vector are highly correlated and have a strong linear relationship.

5. Why is it important for the covariance matrix of a complex random vector to be positive definite?

A positive definite covariance matrix is important because it ensures that the data in the complex random vector are well-behaved and can be used for further analysis. It also allows for the use of various statistical techniques, such as principal component analysis, to be applied to the data.

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