Stochastic Matrices

Go to: Introduction, Notation, Index


In all the expressions below, x is a vector of real or complex random variables with whose mean vector and covariance matrix are given by: <x> = m and Cov(x)=<(x-m)(x-m)H> = S. We define the real-valued vector of variances s=diag(S).

 •,  ÷, •2, ½, abs() and exp() are elementwise operators for multiplication, division, square, square root, absolute value and exponentiation. ⊗ denotes the Kroneker product and j dentotes (-1)½. <Y> denotes the expected value of Y. ||y|| = √(yHy) is the Euclidean vector norm and |Y| is the matrix determinant.

Vectors and matrices a, A, b, B, c, C, d and D are constant (i.e. not dependent on x).

General Properties

Special Distributions

The expressions for cubic and quartic expectations given below are restricted to the following special distributions:


Real Gaussian

Complex Gaussian

Definition: In this section, <=> represents the  Complex-to-Real isomporphism in which we replace each complex element, z, of a complex matrix C by a 2#2 real matrix [zR -zI; zI zR]=|z|×[cos(t) -sin(t); sin(t) cos(t)] where t=arg(z).  <-> represents the corresponding vector mapping in which we replace each complex element, z, of a complex vector c by a 2#1 real vector [zR; zI ].

[x[n]:Complex Gaussian] means that if x[n] <-> y[2n] , then y ~ N(y ; a, ½K) for some complex m[n] <-> a[2n] and +ve definite hermitian S[n#n] <=> K[2n#2n]. In other words, the real and imaginary components of x are jointly gaussian with a symmetric covariance matrix that lies in the range of the complex-to-real isomorphism.

In the two following sections, we define  d=s½=diag(S)½ to be a positive real-valued vector of standard deviations and write |m|=|m| and |S|=|S| for elementwise absolute value functions. The function 1F1(a,b;z)=M(a,b,z) is the Confluent Hypergeometric or Kummer function, hypergeom(a,b,z) in MATLAB. The function 2F1(a,b;z) is the Confluent Hypergeometric function, hypergeom(a,b,z) in MATLAB.

Linear Expectations

For [x: Real Gaussian] :

Quadratic Expectations

For [x[n]: Real Gaussian] :

For [x: Complex Gaussian] :

Minimizing Quadratic Expectations

Cubic Expectations

For [x:Real Independent] :

For [x: Real Gaussian] :

Quartic Expectations

For [x: Real Gaussian] :

For [x: Complex Gaussian] :

Quintic Expressions

For [x: Real Gaussian] :

Recursion Formulae for High Powers

For [x: Real Gaussian] :

For [x: Complex Gaussian] :

Product of Vector Elements

For [x: Real Gaussian] :

For  [x: Complex Gaussian] :

This page is part of The Matrix Reference Manual. Copyright © 1998-2022 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "".
Updated: $Id: expect.html 11291 2021-01-05 18:26:10Z dmb $