By D. J. Hatter

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775 The algorithm for the subdominant eigenvector is, therefore, (1) (2) (3) (4) Form a column vector of order (N- 1) from the first column of [^4] without the leading element. Multiply this vector by -1 and call this, say, {v}. Partition the matrix [A] to give a square matrix of order (N-l) by removing the first row and column. Subtract the first eigenvalue from the diagonal elements of this matrix. Invert the resulting matrix. EIGENVALUES AND EIGENVECTORS (5) (6) 41 Postmultiply the inverted matrix by \V\ The result is the required eigenvector less its leading element of unity.

The subroutine GMPRD in the Scientific Subroutine Package on the IBM 1130 is used for obtaining the product of two compatible matrices. The program CALL is CALL GMPRD (A,B,C,J,K,L) where A is the product matrix B is the premultiplying matrix C is the postmultiplying matrix / is the number of rows in B K is the number of columns in B and number of rows in C L is the number of columns in C Thus if the program requires the matrix multiplication [X] = [Y] x [Z] where Y and Z are both, say, (4 x 4) matrices, the mainline program will include the statement CALL GMPRD (X,Y,Z,4,4,4) Similar subroutines will be available for all the matrix operations and all are usually straightforward to use, although care is needed with inversion routines, which are susceptible to ill-conditioning.

Calderbank, V. 1 INTRODUCTION When an elastic mechanical system is displaced from its equilibrium position and then released, it will oscillate about that position before returning to a state of rest. A car bounced on its springs or the vibrating string of a musical instrument are two examples of this oscillation which is termed mechanical vibration. The phenomenon is present in all mechanical systems, more complex systems giving correspondingly complex vibration characteristics. This chapter will formulate a method of dealing with any linear vibrating system, from the elementary to the complex, and it will show how a general approach capable of analysing the dynamic behaviour of any linear mechanical system may be adopted.