Eigenvector mystery

[Michael de Estrada]

The thing to understand is that the eigenvectors {x} are the solution to the series of simultaneous equations [A]{x} = {B}, where [A] is a square matrix with a zero determinant, such that there are an infinite number of solutions. The eigenvalues are those values in the matrix [A] that result in the determinant being zero. The diagonal coefficients of [A] are modified by subtracting the unknown eigenvalue and solving for those values that result in a zero determinant. The eigenvectors can then be determined by substituting each eigenvalue, and solving for {x} in the homogeneous set of equations [A]{x} = {0}. As has been mentioned before, although there are an infinite number of solutions, they are all the same function shape, differing only by a scale or normalizing factor. This is why eigenvectors are commonly referred to as modeshapes when used to solve problems in vibration theory. They are just as important as eigenvalues when solving problems involving structural vibrations, as they permit the calculation of the distribution of inertial forces acting on a vibrating structure. In fact, the eigenvalues, the square root of which are referred to as natural frequencies, are merely an intermediate step in the computation of the eigenvectors.