# Mean of matrix?

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?

• It's the mean of a vector, which is an entrywise mean. – Sean Roberson Jan 14 at 9:00

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 $$\mu = \frac 1n \left( x_1 + \cdots + x_n\right).$$
• What is $E(\mathbf{X}_i)$)? – noflow Jan 14 at 9:40
• The expectation (AKA the mean) of the random variable $X_i$. – Omnomnomnom Jan 14 at 9:45
• 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 – noflow Jan 14 at 9:59