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If $\mathbf {Z}$ is random vector and $A$ is a fixed matrix, could someone explain why $$\mathrm{cov}[A \mathbf {Z}]= A \mathrm{cov}[\mathbf {Z}]A^\top.$$

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up vote 8 down vote accepted

For a random (column) vector $\mathbf Z$ with mean vector $\mathbf{m} = E[\mathbf{Z}]$, the covariance matrix is defined as $\operatorname{cov}(\mathbf{Z}) = E[(\mathbf{Z}-\mathbf{m})(\mathbf{Z}-\mathbf{m})^T]$. Thus, the covariance matrix of $A\mathbf{Z}$, whose mean vector is $A\mathbf{m}$, is given by $$\begin{align}\operatorname{cov}(A\mathbf{Z}) &= E[(A\mathbf{Z}-A\mathbf{m})(A\mathbf{Z}-A\mathbf{m})^T]\\ &= E[A(\mathbf{Z}-\mathbf{m})(\mathbf{Z}-\mathbf{m})^TA^T]\\ &= AE[(\mathbf{Z}-\mathbf{m})(\mathbf{Z}-\mathbf{m})^T]A^T\\ &= A\operatorname{cov}(\mathbf{Z})A^T. \end{align}$$

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I corrected my typo. Thanks for pointing out my error. –  user92612 Aug 29 at 14:54

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