Singular value decomposition of a 4 x 5 matrix
Consider the matrix
![\[ A = \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 2 \\0 & 0 & 3 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 4 & 0 & 0 & 0 \\\end{array} \right). \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-683802c91644a5828f52492e53a80db6_l3.png "Rendered by QuickLaTeX.com")
A singular value decomposition of this matrix is given by 
, with
![\[ U = \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) , \quad \tilde{\mathit{S}} = \left( \begin{array}{ccccc} 4 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{5} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) , \quad V^T = \left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sqrt{0.2} & 0 & 0 & 0 & \sqrt{0.8} \\ 0 & 0 & 0 & 1 & 0 \\ - \sqrt{0.8} & 0 & 0 & 0 & \sqrt{0.2} \\ \end{array} \right) . \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-eab722bfc814bb62899931e823949450_l3.png "Rendered by QuickLaTeX.com")
Notice above that 
has non-zero values only in its diagonal, and can be written as
![\[ \tilde{\mathit{S}} = \mathbf{diag}(\mathit{S}, 0, 0), \quad \mathit{S} := \mathbf{diag}(\sigma_1,\sigma_2,\sigma_3), \]](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-cd54b01ffe871e9077957fa852a879d3_l3.png "Rendered by QuickLaTeX.com")
with 
. The rank of 
(which is the number of non-zero elements on the diagonal matrix ) is thus 
. We can check that 
, and 
.
