# 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.