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:
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“poor Felix Randal;
How far from then forethought of, all thy more boisterous years,
When thou at the random grim forge, powerful amidst peers,
Didst fettle for the great gray drayhorse his bright and battering
sandal!”
—Gerard Manley Hopkins (1844–1889)