A regularization algorithm for matrices of bilinear and sesquilinear forms

Over a field or skew field FF with an involution a↦a˜ (possibly the identity involution), each singular square matrix A is *congruent to a direct sumS∗AS=B⊕Jn1⊕⋯⊕Jnp,1⩽n1⩽⋯⩽np,inwhich S   is nonsingular and S∗=S∼T; B is nonsingular and is determined by A up to ∗congruence; and the ni-by-ni singular Jordan blocks JniJni 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 F=CF=C (respectively, F=RF=R), 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∗A+λA∗ is decomposed by our regularization algorithm into the direct sumS∗(A+λA∗)S=(B+λB∗)⊕(Jn1+λJn1∗)⊕…⊕(Jnp+λJnp∗)with selfadjoint summands.