A geometric perspective on the singular value decomposition (Q2809909)

The authors start from the characterization of the singular value decomposition (SVD) of a matrix as being given by the best rank \(1\) approximation and from the characterization of the best rank \(r\) approximation. Then, they introduce geometric concepts like secant varieties, their tangent spaces, and critical points of the distant function of a matrix and varieties. This allows for a redefinition and a new proof of the SVD leading also to its coordinate-free version. Continuing, they consider singular vector tuples and singular spaces of tensors and prove a result on (symmetric) tensor decompositions for general (symmetric) tensors based here on earlier work of \textit{S. Friedland} and \textit{G. Ottaviani} [Found. Comput. Math. 14, No. 6, 1209--1242 (2014; Zbl 1326.15036)]. Finally, they give a definition of the Euclidean distance degree and higher-order SVD and formulate generalizations of their former results using these concepts.