Pencils of complex and real symmetric and skew matrices. (English) Zbl 0726.15007
The appearance of this paper in print is a historic event. For the last 18 years the corresponding manuscript has been held in high esteem by most anyone interested in normal forms for symmetric, Hermitian or skew matrix pencil and this reviewer has sent out numerous copies of Thompson’s original 1973 exposition to those that have queried him on the subject. Wherein then lies the value of this paper?
It is an exposition of canonical pair forms that can be achieved under simultaneous congruence for two real symmetric, complex Hermitian, and skew matrices or mixed pairs. It is done much in the style and notation of F. R. Gantmacher’s chapter XII Volume 2 of his book “Applications of the theory of matrices” (1959; Zbl 0085.010, Russian original 1953; Zbl 0050.248). It uses Gantmacher’s minimal indices of a pencil A-\(\rho\) B and, in a way, completes his work. This paper gives explicit constraints on the elementary divisors of the pencil in each possible case. There are essentially two different types of pencils: those that are governed by inertia laws under congruence, e.g. real symmetric ones, and those that are not, e.g. complex symmetric pencils. The question of uniqueness of the parameters of the canonical pair form for inertia governed pencils takes significant work and skill to settle. It is presented here in the shortest way known to the reviewer.
Added to the original 1973 manuscript are a list of references and an introduction to them, making it obvious why so many researcher have needed this information: minimal indices are essential tools in modern control theory; numerical analysis, particularly eigenvalue perturbations can be studied in detail via pencils; the Hasse-Minkowski principle of number theory is intimately related to simultaneous congruence.
The proofs and results of this exposition are not original or new; however, to find all cases combined in one concise paper with a lucid presentation of the underlying ideas is a welcome addition to the pencil literature.
It is an exposition of canonical pair forms that can be achieved under simultaneous congruence for two real symmetric, complex Hermitian, and skew matrices or mixed pairs. It is done much in the style and notation of F. R. Gantmacher’s chapter XII Volume 2 of his book “Applications of the theory of matrices” (1959; Zbl 0085.010, Russian original 1953; Zbl 0050.248). It uses Gantmacher’s minimal indices of a pencil A-\(\rho\) B and, in a way, completes his work. This paper gives explicit constraints on the elementary divisors of the pencil in each possible case. There are essentially two different types of pencils: those that are governed by inertia laws under congruence, e.g. real symmetric ones, and those that are not, e.g. complex symmetric pencils. The question of uniqueness of the parameters of the canonical pair form for inertia governed pencils takes significant work and skill to settle. It is presented here in the shortest way known to the reviewer.
Added to the original 1973 manuscript are a list of references and an introduction to them, making it obvious why so many researcher have needed this information: minimal indices are essential tools in modern control theory; numerical analysis, particularly eigenvalue perturbations can be studied in detail via pencils; the Hasse-Minkowski principle of number theory is intimately related to simultaneous congruence.
The proofs and results of this exposition are not original or new; however, to find all cases combined in one concise paper with a lucid presentation of the underlying ideas is a welcome addition to the pencil literature.
Reviewer: F.Uhlig (Auburn)
MSC:
| 15A22 | Matrix pencils |
| 15A21 | Canonical forms, reductions, classification |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A63 | Quadratic and bilinear forms, inner products |
Keywords:
matrix pencil; Hermitian matrix; symmetric matrix; skew matrix; congruence transformation; canonical form; minimal index; canonical pair formsReferences:
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