In this section we assume that we are dealing with a real Euclidean space
$$ f : E \rightarrow E $$
be any linear map. In general, it may not be possible to diagonalize
. We show that every linear map can be diagonalized if we are willing to use
orthonormal bases. This is the celebrated
singular value decomposition (SVD)
. A close cousin of the SVD is the
of a linear map, which shows how a linear map can be decomposed into its purely rotational component (perhaps with a flip) and its purely stretching part.