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I see that the definition of a covariance matrix uses the mean of another matrix in its definition. However, I cannot find a definition of a mean of a matrix. So how is the mean defined?

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  • $\begingroup$ It's the mean of a vector, which is an entrywise mean. $\endgroup$ – Sean Roberson Jan 14 at 9:00
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In the context of the wiki page that you linked, we have $$ \mu_{\mathbf X} = (E(X_1),E(X_2),\dots,E(X_N))^T. $$


Taking "the mean of a matrix" is not something that makes sense unless we have further information. In a typical application, we would have column-vectors $x_1,\dots,x_n \in \Bbb R^m$ that correspond to distinct measurements. The data matrix corresponding to these measurements would be $$ X = \pmatrix{x_1 & \cdots & x_n} \in \Bbb R^{m \times n} $$ and the mean of the data matrix (which is real the mean of our data set) would be $$ \mu = \frac 1n \left( x_1 + \cdots + x_n\right). $$

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  • $\begingroup$ What is $E(\mathbf{X}_i)$)? $\endgroup$ – noflow Jan 14 at 9:40
  • $\begingroup$ The expectation (AKA the mean) of the random variable $X_i$. $\endgroup$ – Omnomnomnom Jan 14 at 9:45
  • $\begingroup$ But why is $\mathbf{X}_i$ a random variable when my matrix $\mathbf{X}$ is not random? It does not make sense to me. It would make more sense if we were talking about rows/columns. I am talking in the context of covariance of a matrix btw $\endgroup$ – noflow Jan 14 at 9:59
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    $\begingroup$ You need to provide a bit more context to your question; the "covariance of a matrix" only makes sense if the matrix is related to random variables somehow. In your matrix, is every measurement a row, or is every measurement a column? $\endgroup$ – Omnomnomnom Jan 14 at 10:05
  • $\begingroup$ See my latest edit. $\endgroup$ – Omnomnomnom Jan 14 at 10:29

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