Abstract
I MANY PROBLEMS dealing with the eigenvalues and eigenvectors, it is convenient to use a well-known method due to Jacobi for diagonalizing real symmetric matrices. I t consists of performing a sequence of plane rotations (orthogonal transformations), each of which reduces to zero one of the nondiagonal elements of the matrix, until the sum of squares of nondiagonal elements fails to decrease. Less attention has been paid, however, to the total number of rotations required to diagonalize matrices of various orders. Recently, in connection with a study of the effect of the selection of the number of masses and their distribution in approximating the dynamic characteristics of a uniform beam, Jacobi's method was used to calculate the natural frequencies, modes, and other related matrices for a range from 2 to 20 lumped masses using two different methods of lumping (cases a through d) as shown in Fig. 1.
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