We have our design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \). With the SVD we decompose it as $$ \boldsymbol{X} = \boldsymbol{U\Sigma V^T}, $$
with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in {\mathbb{R}}^{n\times p} \) and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).
The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).