Expected value and variance of matrix

Expected value and variance of matrix

There is matrix operation denoted as.

1. Expected value

Proof E(AY)=AE(Y). Suppose A is a constant matrix and Y is a random matrix.

2. Variance

Regarding random variable yi and yj, its variance and covariance are:

Therefore, variance-covariance matrix of a random vector Y:

There is Var[AY]=AVar(Y)A' if A is a constant matrix and Y is a random matrix.