 The fact that "S" = 0 means that "F" is contained in the kernel of "T".Conversely, if "x" ∈ ker("T"), then by self-adjointness, "x" is orthogonal to every eigenvector {"en"} with non-zero eigenvalue.It follows that , and that {"en"} is an orthonormal basis for the orthogonal complement of the kernel of "T".