A Canonical Form for Real Matrices under Orthogonal Transformations.

If A is a square matrix of order n with real or complex elements it is well known that it may be reduced by means of a unitary transformation U to a matrix of the same order all of whose elements below the leading diagonal are zero.' Even when the elements of A are real the elements of the transforming matrix U are complex if the characteristic numbers of A are not all real and it is desirable to give a canonical form which may be reached by the use of real unitary (i.e., orthogonal) matrices. The derivation of this canonical form differs only in detail from that given by Schur. The characteristic numbers X of the matrix A are determined by the equation det(A XE) = 0, where E is the unit matrix, and they may be real or complex. If, as we suppose, the elements of A are real the complex roots will occur in conjugate imaginary pairs. If all the characteristic numbers are real the unitary transformations occurring in Schur's derivation will be real and the canonical form sought for is that given by Schur. On the other hand, let Xi = ,u + iv and X2 = iv be a pair of conjugate complex characteristic numbers of the matrix A (j,, v real, v $ 0); on denoting by xi = a + ib, (a, b, real) a characteristic vector of A associated with the characteristic number Xi we have Ax, = Xix, which implies the two equations