5 Matrix Notation

Some problems in mechanics require for their description: in the two-dimensional case a set of four numbers (or expressions), in the three-dimensional case a set of nine numbers, or, in the nine-dimensional case, a set of 36 numbers. Examples are the orientation or motion of a rigid body in a plane or in space (transformation matrix) or the description of the stress–strain properties of an inhomogeneous, anisotropic material, for example bone (stiffness matrix). To formulate the relevant equations clearly, matrix notation is employed. Use also can be made of established theorems regarding mathematical properties of matrices. The following introduction to matrix notation covers only the arithmetic operations using matrices that are employed later on in this book.

5.1 Definition of a Matrix

Quadratic matrices are sets of four (for the two-dimensional case) or nine numbers or expressions (for the three-dimensional case), ordered in rows and columns. Like the symbolic notation of a three-dimensional vector v in the form of a column

a 2 x 2 matrix M is written

and a 3 x 3 matrix M

The position of each element is designated by two indices. The matrix element M₂₃ (spoken as “M sub two three”) denotes the element in the second row and the third column.

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