A regularization algorithm for matrices of bilinear and sesquilinear forms

Over a field or skew field with an involution (possibly the identity involution), each singular square matrix A is *congruent to a direct sum inwhich S is nonsingular and ; B is nonsingular and is determined by A up to *congruence; and the ni-by-ni singular Jordan blocks Jni and their multiplicities are uniquely determined by A. We give a regularization algorithm that needs only elementary row operations to construct such a decomposition. If (respectively, ), we exhibit a regularization algorithm that uses only unitary (respectively, real orthogonal) transformations and a reduced form that can be achieved via a unitary *congruence or congruence (respectively, a real orthogonal congruence). The selfadjoint matrix pencil A+λA* is decomposed by our regularization algorithm into the direct sum with selfadjoint summands.