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 propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)
“There is a potential 4-6 percentage point net gain for the President [George Bush] by replacing Dan Quayle on the ticket with someone of neutral stature.”
—Mary Matalin, U.S. Republican political advisor, author, and James Carville b. 1946, U.S. Democratic political advisor, author. All’s Fair: Love, War, and Running for President, p. 205, Random House (1994)