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Reduction to tridiagonal form and minimal realizations. (English) Zbl 0754.65040

The paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix.

In the preparative part of the paper the author describes the representation of the class of similar tridiagonals by vector pairs and the use of a pair $\left(\stackrel{^}{T},{\Omega }\right)$, with $\stackrel{^}{T}$ symmetric tridiagonal and ${\Omega }$ diagonal, rather than a single matrix ${{\Omega }}^{-1}T$.

The fundamental result of the paper says that the tridiagonal reduction is equivalent to a Gram-Schmidt process applied to two Krylow sequences. In Euclidean space a proper normalization allows one to monitor a tight lower bound on the condition number of the transformation.

##### MSC:
 65F30 Other matrix algorithms 65F15 Eigenvalues, eigenvectors (numerical linear algebra) 65K10 Optimization techniques (numerical methods) 15A21 Canonical forms, reductions, classification 93B10 Canonical structure of systems