Numerical Calculations in Clifford Algebra

13Matrices of Clifford Numbers

Systems of linear equations in which the coefficients¹ are Clifford entities of a particular type, can be solved by matrix inversion when the inverses of the coefficients (elements of the matrix) can be calculated, and when the inverses and products of those coefficients yield entities of the same type. This is the case when the Clifford entities are scalars, bicomplex numbers, quaternions, Pauli matrices, and extended electromagnetic fields.

Any inversion technique which applies to scalar coefficients (these being commutative) can be extended to coefficients of higher grades, as long as care is taken to honour non‐commutative multiplication. As a practical example, the inversion of systems with extended electromagnetic fields as coefficients in the context of electromagnetic scattering is examined in Chapter 25. Here the fundamentals are demonstrated by applying, for simplicity, Gaussian elimination (Grcar 2011) on smaller systems involving bicomplex numbers.

13.1 Background

Matrices are accessed within the Clifford numerical suite using the special data type ‘matrix’. The data type is defined as the structure shown in Example 13.1.

Example 13.1

 1: /* matrix type definition */
 2: typedef struct{
 3:   int nrows;        /* number of rows */
 4:   int ncols;        /* number of columns */
 5:   int stride;       /* offset between 1st elements in adjacent columns */
 6:   number *data;     /* first element of data */
 7: } matrix;
 

Routines are provided for simple matrix ...