Covariance Matrix - Complex Random Vectors

Complex Random Vectors

The variance of a complex scalar-valued random variable with expected value μ is conventionally defined using complex conjugation:


operatorname{var}(z)
=
operatorname{E}
left[ (z-mu)(z-mu)^{*}
right]

where the complex conjugate of a complex number is denoted ; thus the variance of a complex number is a real number.

If is a column-vector of complex-valued random variables, then the conjugate transpose is formed by both transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix, as its expectation:


operatorname{E}
left[ (Z-mu)(Z-mu)^dagger
right] ,

where denotes the conjugate transpose, which is applicable to the scalar case since the transpose of a scalar is still a scalar. The matrix so obtained will be Hermitian positive-semidefinite, with real numbers in the main diagonal and complex numbers off-diagonal.

Read more about this topic:  Covariance Matrix

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