the (euclideaninner product of and (viewed as column vectorsis now defined as hxyi hermitian unitary xi yi = which is no longer symmetrichxyi hyxi note that this generalizes the real-valued inner product the determinant of complex matrix is defined exactly as in ( as consequencedet(adet(aa complex matrix is said to be hermitian or self-adjoint if aaand unitary if aa (that isif aa- for real matrices "hermitianis the same as "symmetric"and "unitaryis the same as "orthogonala orthogonal projections let { uk be set of linearly independent vectors in rn the set span { uk { ak uk ak }linear subspace orthogonal complement orthogonal projection matrix is called the linear subspace spanned by { uk the orthogonal complement of vdenoted by is the set of all vectors that are orthogonal to vin the sense that hwvi for all the matrix such that px xfor all vand px for all is called the orthogonal projection matrix onto suppose that [ uk has full rankin which case uu is an invertible matrix the orthogonal projection matrix onto span { uk is then given by (uu)- unamelysince pu uthe matrix projects any vector in onto itself moreoverp projects any vector in onto the zero vector using the pseudo-inverseit is possible to specify the projection matrix also for the case where is not of full rankleading to the following theorem theorem orthogonal projection let [ uk thenthe orthogonal projection matrix onto span{ uk is given by ( where uis the (rightpseudo-inverse of proofby property of definition we have pu uuu uso that projects any vector in onto itself moreoverp projects any vector in onto the zero vector note that in the special case where uk above form an orthonormal basis of vthen the projection onto is very simple to describenamely we have px uu = hxui ui (