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]](http://upload.wikimedia.org/math/a/e/d/aedf356d1e06c9057edbee3a8609c55c.png)
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] ,](http://upload.wikimedia.org/math/5/e/1/5e185a012f91dfbc5480f0ad30fb5d8a.png)
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
Famous quotes containing the words complex and/or random:
“When distant and unfamiliar and complex things are communicated to great masses of people, the truth suffers a considerable and often a radical distortion. The complex is made over into the simple, the hypothetical into the dogmatic, and the relative into an absolute.”
—Walter Lippmann (1889–1974)
“It is a secret from nobody that the famous random event is most likely to arise from those parts of the world where the old adage “There is no alternative to victory” retains a high degree of plausibility.”
—Hannah Arendt (1906–1975)