Low Grade Matrices and Matrix Fraction Representations
نویسندگان
چکیده
The lgrade of a n × n matrix A is the largest rank of any subdiagonal block of a symmetric partition of A. A number of algebraic results on lgrade are given. When A has lgrade d, it can be be approximately decomposed as A = U + V , where U is an upper triangular matrix and V has rank d. If A satisfies GA = N with G and N have lower bandwidths dG and dN , then the decomposition is exact: A = U + V , where U is an upper triangular matrix with lower bandwidth equal to dN − dG and V has low rank (generically dG). This result generalizes well known representations of A when A = G−1 and G is banded. A generalization of the Givens rotation product decomposition of unitary Hessenberg matrices is given and its structure analyzed. These ‘consecutive subblock products” are used to construct a representations of a lgrade-d matrix A of the form: GA = N with G and N having lower bandwidth d. G can be chosen to be lower triangular or unitary. AMS Classifications: 15A23
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