As an example, consider the following 3\times 2 example for the matrix \boldsymbol{\Sigma}
\boldsymbol{\Sigma}= \begin{bmatrix} 2& 0 \\ 0 & 1 \\ 0 & 0 \\ \end{bmatrix}The singular values are \sigma_0=2 and \sigma_1=1 . It is common to rewrite the matrix \boldsymbol{\Sigma} as
\boldsymbol{\Sigma}= \begin{bmatrix} \boldsymbol{\tilde{\Sigma}}\\ \boldsymbol{0}\\ \end{bmatrix},where
\boldsymbol{\tilde{\Sigma}}= \begin{bmatrix} 2& 0 \\ 0 & 1 \\ \end{bmatrix},contains only the singular values. Note also (and we will use this below) that
\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}= \begin{bmatrix} 4& 0 \\ 0 & 1 \\ \end{bmatrix},which is a 2\times 2 matrix while
\boldsymbol{\Sigma}\boldsymbol{\Sigma}^T= \begin{bmatrix} 4& 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0\\ \end{bmatrix},is a 3\times 3 matrix. The last row and column of this last matrix contain only zeros. This will have important consequences for our SVD decomposition of the design matrix.