This paper is concerned with the determination of the Jordan canonical form and D1/,2-norm of the SOR iterative matrix derived from the coefficient matrix A having the form *-& t) with D\ and Di symmetric and positive definite. The theoretical results show that the Jordan form is not diagonal, but has only q principal vectors of grade 2 and that the D1//2-norm of J2?Ub (u^, the optimum paramete...

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....

In mathematics, complete classification of structures, such as groups and rings, is often a primary goal. Linear transformations are no exception to this. Certain canonical forms exist to classify linear transformations, therefore creating a unique representative of linear transformations in the same similarity class. Diagonal representation is of course one of the simplest examples of a canoni...

For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, th...

We show how Ptak’s duality method leads to short proofs of two extensions of the Jordan canonical form, viz. the normal form for a matrix over an arbitrary (not necessarily algebraically closed) field under similarity and the canonical form for a pair of matrices under contragredient equivalence. The duality method is summarized in the following. Lemma. Let V be a finite-dimensional space over ...

An algebraic classification of second order symmetric tensors in 5-dimensional Kaluza-Klein-type Lorentzian spaces is presented by using Jordan matrices. We show that the possible Segre types are [1, 1111], [2111], [311], [z z̄ 111], and the degeneracies thereof. A set of canonical forms for each Segre type is found. The possible continuous groups of symmetry for each canonical form are also stu...

a common method to determine the order of minimal realization of a continuous linear time invariant descriptor system is to decompose it into slow and fast subsystems using the weierstrass canonical form. the weierstrass decomposition should be avoided because it is generally an ill-conditioned problem that requires many complex calculations especially for high-dimensional systems. the present ...