therefore det( and hence that is an eigenvalue of alet be an eigenvector corresponding to thenaw lw orequivalentlywa lwfor this reasonwe call wthe left-eigenvector of for eigenvalue if is (right-eigenvector of athen its adjoint vis usually not left-eigenvectorunless aa aa(such matrices are called normala real symmetric matrix is normalhoweverthe important property holds that leftand right-eigenvectors belonging to different eigenvalues are orthogonal namelyif wis left-eigenvalue of and right-eigenvalue of then wv wav wvwhich can only be true if wv theorem schur triangulation for any complex matrix athere exists unitary matrix such that - au is upper triangular proofthe proof is by induction on the dimension of the matrix clearlythe statement is true for as is simply complex number and we can take equal to suppose that the result is true for dimension we wish to show that it also holds for dimension any matrix always has at least one eigenvalue with eigenvector vnormalized to have length let be any unitary matrix whose first column is such matrix can always be constructed as is unitarythe first row of - is vand - au is of the form { for some matrix by the induction hypothesisthere exists unitary matrix and an upper triangular matrix such that - bw nowdefine : then## - au - - bw - which is upper triangular of dimension since uv is unitarythis completes the inductionand hence the result is true for all the theorem above can be used to prove an important property of hermitian matricesi matrices for which aa after specifying we can complete the rest of the unitary matrix via the gram-schmidt procedurefor examplesee section lefteigenvector normal matrix