
If the p × 1 random vector X has variancecovariance matrix Σ and A is an m × p matrix of constants, prove that the variancecovariance matrix of AX is AΣA′. Start with the definition of a variancecovariance matrix:
cov(Z) = E(Z − μz)(Z − μz)′.
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`sum=cov(X)=E(Xmux)(Xmux)'`
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(AXAmux)(AXAmux)'`
`=E(A(Xmux))(A(Xmux))'`
`=AE(Xmux)(Xmux)'A'`
( A is constant and by reversal law of transpose matrix)
`=A(E(Xmux)(Xmux)')A'`
(A is constant, so A' is also constant and property of E )
`cov(AX)=AsumA'`
Since
`cov(X)=E(Xmux)(Xmx)'=sum`
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