Scaling of the covariance matrix upon multiplication with constant matrix

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Scaling upon multiplication with matrix

Theorem: The covariance matrix $\Sigma_{XX}$ of a random vector $X$ scales upon multiplication with a constant matrix $A$:

\[\label{eq:covmat-scal} \Sigma(AX) = A \, \Sigma(X) A^\mathrm{T} \; .\]

Proof: The covariance matrix of $X$ can be expressed in terms of expected values as follows:

\[\label{eq:covmat} \Sigma_{XX} = \Sigma(X) = \mathrm{E}\left[ (X-\mathrm{E}[X]) (X-\mathrm{E}[X])^\mathrm{T} \right] \; .\]

Using this and the linearity of the expected value, we can derive \eqref{eq:covmat-scal} as follows:

\[\label{eq:covmat-scal-qed} \begin{split} \Sigma(AX) &\overset{\eqref{eq:covmat}}{=} \mathrm{E}\left[ ([AX]-\mathrm{E}[AX]) ([AX]-\mathrm{E}[AX])^\mathrm{T} \right] \\ &= \mathrm{E}\left[ (A[X-\mathrm{E}[X]]) (A[X-\mathrm{E}[X]])^\mathrm{T} \right] \\ &= \mathrm{E}\left[ A (X-\mathrm{E}[X]) (X-\mathrm{E}[X])^\mathrm{T} A^\mathrm{T} \right] \\ &= A \, \mathrm{E}\left[ (X-\mathrm{E}[X]) (X-\mathrm{E}[X])^\mathrm{T} \right] A^\mathrm{T} \\ &\overset{\eqref{eq:covmat}}{=} A \, \Sigma(X) A^\mathrm{T} \; . \end{split}\]

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Sources:

Wikipedia (2022): "Covariance matrix"; in: Wikipedia, the free encyclopedia, retrieved on 2022-09-22; URL: https://en.wikipedia.org/wiki/Covariance_matrix#Basic_properties.

Metadata: ID: P348 | shortcut: covmat-scal | author: JoramSoch | date: 2022-09-22, 11:45.