# Spatial and temporal covariance matrices

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 ?

For a meaningful answer to your question about "physical interpretation", I need to work with empirical covariance matrices, so averaged over many trials. (You consider a single trial in your question, but that somehow obscures the interpretation.) My conclusion is:

If you find that the empirical spatial and temporal covariance matrices share the same positive eigenvalues, then you know that your data is self-averaging in both space and time.

Consider a data set $x_{ij}^{(k)}$ of measured values at position $i$, time sample $j$ in the $k$-th trial. The empirical spatio-temporal covariance matrix is defined by

$$C_{ii',jj'}=\frac{1}{N_{\rm trials}}\sum_{k=1}^{N_{\rm trials}}x_{ij}^{(k)}x_{i'j'}^{(k)}.$$

Empirical spatial and temporal covariance matrices are constructed as partial traces,

$$S_{ii'}=\sum_{j}C_{ii',jj},\qquad T_{jj'}=\sum_{i}C_{ii,jj'}.$$

In general, these two matrices $S$ and $T$ will have different sets of positive eigenvalues.

However, if the data is self-averaging in time, then the empirical spatial covariance matrix is $\bar{S}_{ii'}=\sum_{j}x_{ij}^{(k)}x_{i'j}^{(k)}$ independent of the sample index $k$; if moreover the data is self-averaging in space, then the empirical temporal covariance matrix is $\bar{T}_{jj'}=\sum_{i}x_{ij}^{(k)}x_{ij'}^{(k)}$ independent of $k$. These two matrices $\bar{S}$ and $\bar{T}$ (corresponding to your $A$ and $B$) have the same set of positive eigenvalues.