A note on the representation and definition of semiseparable matrices

In this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semisepar-able matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition. Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitely formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation. As an example of the effectivity of the new representation, we design on O(n) algorithm for the multiplication of a semiseparable matrix given by the new representation, with a vector. A note on the representation and definition of semiseparable matrices * Abstract In this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semiseparable matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition. Because of the alternative definition, the standard representation of semisepar-able matrices is not satisfying anymore. The concept of a representation is expli-citely formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation. As an example of the effectivity of the new representation, we design on O(n) algorithm for the multiplication of a semiseparable matrix given by the new representation , with a vector.