Suppose $(x_i(t))$ is a $n$-dimensional time-series, where $t$ is an integer between $1$ and $T$ (time is discrete) and $i$ an integer between $1$ and $n$, and I assume $n<T$. From this time-series, one can construct two interesting covariance matrices:

The spatial covariance matrix : $A_{ij} = \sum_{t} x_i(t)x_j(t)$

The temporal covariance matrix : $B_{st} = \sum_{i} x_i(s)x_i(t)$

If one puts $x_i(t)$ in matrix form $X$, with $X_{it}=x_i(t)$, then $A=X.X'$ and $B=X'.X$.

So from the singular value decomposition of $X$ we know that $A$ and $B$ share exactly the same eigenvalues (the only difference is that, in addition, $B$ has $T-n$ zero eigenvalues).

Therefore, my question is : what does it MEAN ? Is there an explanation of this fact somewhere ? Do you have some physical interpretation ?