### On the Gyration of a Matrix

J.A. Stark and W.J. Fitzgerald

Technical Report CUED/F-INFENG/TR 229 (1995), Department of Engineering, University of Cambridge

An operator termed the gyration of a matrix is investigated. This operator is developed from the concept of inverting parts of matrices; the result contains the inverse of a submatrix of the operand matrix. The gyration is defined in terms of fundamental conditions for consistent behaviour, and these are shown to determine its algebraic form. A brief survey of the properties of the gyration is presented. A variety of matrix methods cush as those of numerical inversion can be conveniently expressed using the gyrationn formalism.

We discuss the application of the gyration to statistics and statistical data modelling, and this is followed by more detailed study of the problem of searching for the optimal data model. A central theme in this discussion is the use of matrix augmentation; many sum-of-product matrix expressions can be conveniently formulated as part of the gyration of augmented matrices. The properties that follow from the consistency conditions can be used to manipulate the resulting equations.

The matrix gyration operator can be generalised with an additional parameter specifying the degree or fraction to which the matrix is gyrated. This amounts to inverting parts of a matrix by variable amounts. The fractional gyration is shown to possess the same fundamental consistency properties of the standard form. An application in statistics is suggested.