Given a matrix $\mathbf X \to X_{m}^{\phantom{m}n}$, we can decompose it into three matrices

$$X_{m}^{\phantom{m}n} = U_{m}^{\phantom{m}k} D_{k}^{\phantom{k}l} (V_{n}^{\phantom{n}l} )^{\mathrm T},$$

where $D_{k}^{\phantom{k}l}$ is diagonal.

Here we have $\mathbf U$ being constructed by the eigenvectors of $\mathbf X \mathbf X^{\mathrm T}$, while $\mathbf V$ is being constructed by the eigenvectors of $\mathbf X^{\mathrm T} \mathbf X$ (which is also the reason we keep the transpose).

I find this slide from Christoph Freudenthaler very useful. The original slide has been added as a reference to this article.

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