Given a random variable $X \in R$ with zero mean and covariance matrix $\Sigma$. What is the mean of $Z=X^{T}X$?
The answer is $trace({\Sigma})$. What is $trace$ and how do you compute this?
2026-04-01 04:39:19.1775018359
Given a random variable $X \in R$ with zero mean and covariance matrix $\Sigma$. What is the mean of $Z=X^{T}X$?
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The trace of an $n$-by-$n$ square matrix $A$ is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of $A$, i.e $$ \operatorname{tr}(A) = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \dots + a_{nn}. $$ We can write down what $\operatorname E[X^TX]$ looks like and we will see that it is actually the trace of the covariance matrix $\Sigma$.