Algebraic subjects/theory of matrices, and applications

• Main Author: Afriat, S. N.
• Published: University of Oxford 1953
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.729039

This thesis consists of six independent parts, each of which develops a topic in the theory of matrices. The first part, 'Compounds, adjugates and partitioned determinants', attempts, with the introduction of suitable notation, and the proving of some general results, to contribute to and complete certain aspects of the general theory of determinants. It is specially concerned with the derived systems of matrices, the compounds and adjugates of a matrix, whose elements are formed from the minors and cofactors of that matrix, with the dual hybrid compounds, which form a generalization of these, and their determinants, and also with the expansions of bordered and partitioned determinants in terms of the compounds and adjugates of their component parts. The multiplication theorem for compounds and adjugates of Binet and Cauchy is proved, and its application is fundamental throughout. The second part, 'Commutative matrices, latent vectors and characteristic values', discusses commutative sets of matrices, their common latent vectors, and the sets of characteristic values which correspond to them. Also it makes a study of commutative matrix algebras with a unit over the complex field. A set of characteristic values all of which correspond to a single common latent vector of a set of commutative matrices defines a characteristic set of that matrix set. Two fundamental results proved are the that every commutative matrix set has at least one characteristic set, and that the characteristic sets of any commutative matrix set can be extended to give the characteristic sets of any commutative extension of that set. Also it is proved that the matrices of any commutative set can all be reduced to triangular form by the same unitary transformation, and the sets of similarly situated diagonal elements in the triangular transforms, are the characteristic sets with certain repetitions . Through these repetitions the multiplicity of a characteristic set is defined so as to generalize the concept of the multiplicity of a characteristic value of a single matrix as its multiplicity as a root of the characteristic equation. The rank of a characteristic set is defined as the rank of the space of corresponding simultaneous latent vectors of the matrices of the set, the corresponding axial space. It is shown that the rank of a characteristic set is at most its multiplicity. The index of a characteristic set is defined so as to generalize the concept of the index of a characteristic value of a single matrix as its multiplicity as a root of the minimum function, and it it shown that no characteristic set has zero index. Proof is given to the theorem that the characteristic values of any rational function of commutative matrices are given by the values of that function when the argument ranges over the characteristic sets, and the multiplicity of any characteristic value is determined as the sum of the multiplicities of the characteristic sets which give that value.