The analysis of systems having finite freedom is found to require manipulation of linear algebraic equations. These equations are clumsy in form, particularly if the system to which they relate has more than three or four degrees of freedom, so that some form of algebraic shorthand becomes desirable. Matrix notation serves this purpose. The essence of matrix methods is orderliness, and this feature makes the techniques particularly helpful in the setting out of problems for numerical solution by digital computers or by semi-skilled human computers.

The theme of this presentation of vibration theory is that sets of linear equations may be written concisely in matrix form, by the use of certain conventions. The matrices can then be manipulated (for instance to give other matrices) according to certain rules, re-interpretation at any stage being possible through the conventions. The purpose of this first chapter is merely to provide a background in matrix algebra before the treatment of the dynamical problems is presented.

Matrix notation and preliminary definitions

A matrix is an array of numbers (or ‘elements’), the positions as well as the magnitudes of which have significance. It is thus like a determinant, but with one essential difference.