If the p × 1 random vector X has variance-covariance matrix Σ and A is an m × p matrix of constants, prove that the variance-covariance matrix of AX is AΣA′. Start with the definition of a...

  1. If the p × 1 random vector X has variance-covariance matrix Σ and A is an m × p matrix of constants, prove that the variance-covariance matrix of AX is AΣA′. Start with the definition of a variance-covariance matrix:

cov(Z) = E(Z − μz)(Z − μz)′.

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`sum=cov(X)=E(X-mux)(X-mux)'` 

Where `mux` is mean of X, X is matrix of order (p x 1). A is constant matrix of order ( m x p).

`cov(AX)=E(AX-Amux)(AX-Amux)'`

`=E(A(X-mux))(A(X-mux))'`

`=AE(X-mux)(X-mux)'A'` 

( A is constant and by reversal law of transpose matrix)

`=A(E(X-mux)(X-mux)')A'`

(A is  constant, so A' is also constant and property of E )

`cov(AX)=AsumA'`    

Since

 `cov(X)=E(X-mux)(X-mx)'=sum`

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