A regularization algorithm for matrices of bilinear and sesquilinear forms
نویسنده
چکیده
Over a field or skew field F with an involution a 7→ ã (possibly the identity involution), each singular square matrix A is *congruent to a direct sum S∗AS = B ⊕ Jn1 ⊕ · · · ⊕ Jnp , 1 ≤ n1 ≤ · · · ≤ np, in which S is nonsingular and S∗ = S̃ ; 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 F = C (respectively, F = 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∗ is decomposed by our regularization algorithm into the direct sum S∗(A+ λA∗)S = (B + λB∗)⊕ (Jn1 + λJ ∗ n1 )⊕ · · · ⊕ (Jnp + λJ ∗ np ) with selfdajoint summands. AMS classification: 15A63; 15A21; 15A22
منابع مشابه
Canonical matrices of bilinear and sesquilinear forms
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