Describe in a sentence to perform Singular Value Decomposition (SVD) for all $t$.

Question

This question might be more of a linguistic thing than a pure mathematical question. However, I am struggling with writing something down in a proper manner.

I'm trying to describe that I want to perform SVD in each month on a matrix $X_t$ that changes every month (where $t$ indicates the month), thereby obtaining $V^T_t$ and $\Sigma_t$ for all months in my sample. I will then use $V^T_t$ and $\Sigma_t$ for Principal Component Analysis.

Currently, I am writing this down as follows.

Once the data is finalized, SVD is performed in each month independently on the data matrix $X_t$. This results in a component loadings matrix $V^T_t$ for each month, as well as the eigenvalues in $\Sigma_t$. The matrices $V^T_t$ and $\Sigma_t$ will subsequently be employed in reducing the dimensions of the data by means of equation 3.

Where equation 3 simply shows how the dimensionality of the data can be reduced by using the loadings in $V^T_t$.

However, I am unsure whether this is the correct way of writing down that I obtain these matrices in every month. I'm not a mathematician, so would love to get a better grasp of how this should be written down.

Answer 1

How about this?

... SVD is performed independently on the data matrix $X_t$ for each month $t$. This results in a component loadings matrix $V^T_t$ and eigenvalues $\Sigma_t$ for each $t$. The matrices $V^T_t$ and $\Sigma_t$ will subsequently be employed in reducing the dimensions of the data by means of equation 3.