**Properties:**

* Where $U$ is constructed via a full SVD, $U^\top U = I_{p\times p}$ and $U U^\top = I_{m \times m}$

* Where $\hat U$ is constructed via a reduced SVD, although $\hat U^\top \hat U = I_{p\times p}$ it happens that $\hat U \hat U^\top \neq I_{m \times m}$

* Where $\hat U$ is constructed via a reduced SVD, although $\hat U^\top \hat U = I_{p\times p}$, it happens that $\hat U \hat U^\top \neq I_{m \times m}$

We illustrate these properties for our example with the following code cells.

* ${V_k}^{T}=\tilde{\epsilon_k}$ (the $k$th principal component)

Because there are alternative algorithms for computing $P$ and $U$ for given a data matrix $X$, depending on algorithms used, we might have sign differences or different orders between eigenvectors.

Because there are alternative algorithms for computing $P$ and $U$ for given a data matrix $X$, depending on algorithms used, we might have sign differences or different orders of eigenvectors.

We can resolve such ambiguities about $U$ and $P$ by