Invariance of the covariance matrix under addition of constant vector

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Invariance under addition of vector

Theorem: The covariance matrix $\Sigma_{XX}$ of a random vector $X$ is invariant under addition of a constant vector $a$:

\[\label{eq:covmat-inv} \Sigma(X+a) = \Sigma(X) \; .\]

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-inv} as follows:

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

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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: P347 | shortcut: covmat-inv | author: JoramSoch | date: 2022-09-22, 11:29.