Do We Really Need Eigen Values?

Many problems in mathematics and physics reduce "eigenvalue problems", and much effort is expended trying to determine the eigenvalues of a general NxN system, which is sometimes depends on finding the roots of the Nth degree characteristic polynomial. The task of efficiently finding all the complex roots of a high-degree polynomial is non-trivial, and is complicated by the need to consider various special cases such as repeated roots. However, depending on what we're really trying to accomplish, we may not need to find the eigenvalues at all.

The usefulness of the eigenvalues is mainly due to the fact that descriptions of an action are usually simplest when its components are resolved along the directions of the eigenvectors. For example, the eigenvectors of a rotation matrix define its axis of rotation. As a result, it's easy to write equations for the positions of points on a rotating sphere if we work in coordinates that are aligned with the eigenvectors of the rotation, whereas if we work in some other arbitrary coordinates the equations describing the motion will be more complicated.

But there are many situations in which the characteristic axes of the phenomenon can be exploited without ever explicitly determining the eigenvalues. For example, suppose we have the system of three continuous variables x(t), y(t), and z(t) that satisfy the equations

where primes denote derivatives with respect to t, and we want to determine the exact root-matched recurrence formulas for x(nT), y(nT), and z(nT) for a given discrete time interval T. The characteristic polynomial is

where

The root-matched recurrence coefficients are proportional to the elementary symmetric functions (with alternating signs) of the quantities exp(r_kT), where r_k, k = 1, 2, 3 are the eigenvalues, so a common approach is to solve the characteristic equation for the eigenvalues and then compute the recurrence coefficients. Even for a simple cubic the precise determination of the (complex) roots in embedded code, covering all possible cases, is not entirely trivial, and for higher order systems it's even more complicated.

But we don't really need to solve for the eigenvalues. Let w(k) denote the sum of the kth powers of the roots r₁, r₂, r₃. Clearly w(0) = 3, and the values of w(k) for k = 1, 2,... are given by

and the recurrence

for all k > N−1. Now let E(k) denote the sum of the kth powers of the values exp(r_kT), k = 1, 2, 3. Expanding each of the exponential functions as a Taylor series and combining terms by powers of T gives

Letting σ_k denote (-1)^k times the kth elementary symmetric function of the quantities exp(r_jT), we can now compute σ₀, σ₁, σ₂, and σ₃ using the identities

Then the exact homogeneous recurrence formula for x(t) is

and the recurrences for y(t) and z(t) are identical. The summation for E(k) is quite well-behaved and can be implemented quite easily in embedded code. This method, which is directly applicable to systems of any order, requires no root-finding algorithm and no special consideration of repeated roots, etc.

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