Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Expectation of a trace

Theorem: Let $A$ be an $n \times n$ random matrix. Then, the expectation of the trace of $A$ is equal to the trace of the expectation of $A$:

\[\label{eq:mean-tr} \mathrm{E}\left[ \mathrm{tr}(A) \right] = \mathrm{tr}\left( \mathrm{E}[A] \right) \; .\]

Proof: The trace of an $n \times n$ matrix $A$ is defined as:

\[\label{eq:tr} \mathrm{tr}(A) = \sum_{i=1}^{n} a_{ii} \; .\]

Using this definition of the trace, the linearity of the expected value and the expected value of a random matrix, we have:

\[\label{eq:mean-tr-qed} \begin{split} \mathrm{E}\left[ \mathrm{tr}(A) \right] &= \mathrm{E}\left[ \sum_{i=1}^{n} a_{ii} \right] \\ &= \sum_{i=1}^{n} \mathrm{E}\left[ a_{ii} \right] \\ &= \mathrm{tr}\left( \left[ \begin{matrix} \mathrm{E}[a_{11}] & \ldots & \mathrm{E}[a_{1n}] \\ \vdots & \ddots & \vdots \\ \mathrm{E}[a_{n1}] & \ldots & \mathrm{E}[a_{nn}] \end{matrix} \right] \right) \\ &= \mathrm{tr}\left( \mathrm{E}[A] \right) \; . \end{split}\]
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Metadata: ID: P298 | shortcut: mean-tr | author: JoramSoch | date: 2021-12-07, 09:03.