Singular value decomposition in bra-ket notation

In linear algebra, and therefore quantum information, the singular value decomposition (SVD) is elementary, ubiquitous, and beautiful. However, I only recently realized that its expression in bra-ket notation is very elegant. The SVD is equivalent to the statement that any operator $\hat{M}$ can be expressed as

(1) $\begin{align*} \hat{M} = \sum_i \vert A_i \rangle \lambda_i \langle B_i \vert \end{align*}$

where $\vert A_i \rangle$ and $\vert B_i \rangle$ are orthonormal sets of vectors, possibly in Hilbert spaces with different dimensionality, and the $\lambda_i \ge 0$ are the singular values.