definition moore-penrose pseudo-inverse moore-penrose pseudo-inverse the moore-penrose pseudo-inverse of real matrix rnxp is defined as the unique matrix ar pxn that satisfies the conditions aaa aaaa (aa)aa (aa)aa we can write aexplicitly in terms of when has full column or row rank for examplewe always have aaaa(aa)((aa) )( )aleft pseudo-inverse ( if has full column rank pthen (aa)- existsso that from ( it follows that (aa)- athis is referred to as the left pseudo-inverseas aa similarlyif has full row rank nthat is(aa)- existsthen it follows from aaa(aa) ( (aa))aright pseudo-inverse that aa(aa)- this is the right pseudo-inverseas aain finallyif is of full rank and squarethen aa- inner product inner product the (euclideaninner product of two real vectors [ xn ]and [ yn ]is defined as the number hxyi xi yi xy = here is the matrix multiplication of the ( nmatrix xand the ( matrix the inner product induces geometry on the linear space rn allowing for the definition of lengthangleand so on the inner product satisfies the following properties hax byzi hxzi hyzi hxyi hyxi hxxi hxxi if and only if orthogonal euclidean norm vectors and are called perpendicular (or orthogonalif hxyi the euclidean norm (or lengthof vector is defined as || | xn hxxi