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Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2022-09-22 04:45:00 -0700 |
Scaling of the covariance matrix upon multiplication with constant matrix |
General Theorems |
Probability theory |
Covariance |
Scaling upon multiplication with matrix |
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Wikipedia |
2022 |
Covariance matrix |
Wikipedia, the free encyclopedia |
retrieved on 2022-09-22 |
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P348 |
covmat-scal |
JoramSoch |
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:
$$ \label{eq:covmat-scal-qed}
\begin{split}
\Sigma(AX) &\overset{\eqref{eq:covmat}}{=} \mathrm{E}\left[ ([AX]-\mathrm{E}[AX]) ([AX]-\mathrm{E}[AX])^\mathrm{T} \right] \\
&= \mathrm{E}\left[ (A[X-\mathrm{E}[X]]) (A[X-\mathrm{E}[X]])^\mathrm{T} \right] \\
&= \mathrm{E}\left[ A (X-\mathrm{E}[X]) (X-\mathrm{E}[X])^\mathrm{T} A^\mathrm{T} \right] \\
&= A , \mathrm{E}\left[ (X-\mathrm{E}[X]) (X-\mathrm{E}[X])^\mathrm{T} \right] A^\mathrm{T} \\
&\overset{\eqref{eq:covmat}}{=} A , \Sigma(X) A^\mathrm{T} ; .
\end{split}
$$