Applications

In the following three subsections, we will discuss some of the most important special cases and applications of standard operators and linear maps, generally. We start with orthogonal operators, isometries in real vector spaces, including reflections. Next, we return to linear maps, \({{\,\textrm{Hom}\,}}(V,V')\) , on inner product vector spaces and explain in some detail the role of singular value decomposition (SVD). This leads smoothly to the polar decomposition operators of square matrices. We finally discuss the Sylvester’s law of inertia and investigate shortly its connection with special relativity.