Avoided level crossings

In applications, there is not only a requirement for eigenvalues and eigenvectors of matrices; beyond that it is often required to have the eigenvalues of combinations of matrices, such as their sums or products. These quantites are evidently related to those of the constituents, but not always in a way which is easily seen. It is still possible to work out some guidelines.

Consider two matrices $A$ and $B$, normal if you will, and their convex linear combination, by which we mean, $C = ( p A + (1 - p) B )$ for a parameter $p$ varying between zero and one. If the two matrices commute there is no problem: there is a coordinate system in which they are simultaneously diagonal, the eigenvalues $a_i$ and $b_i$ can be listed in order so that we know which pair attach to the same eigenvector, and the new eigenvalues are

$$\begin{eqnarray*} c_i & = & p a_i + (1 - p) b_i. \end{eqnarray*}$$

Of course, indexed order is not necessarily numerical order, so there could be some values of $p$ for which $c_i(p) = c_j(p)$, a degeneracy which is always interesting and sometimes causes numerical problems.

Figure: Eigenvalue crossing for common eigenvectors.

If $A$ and $B$ are changed ever so little, they may no longer commute, changing this picture, although the indexing scheme may still persist. For some insight into what may happen, consider that $A$ is a $2x2$ diagonal matrix with eigenvalues 1 and -1, and that $B$ is skewdiagonal, but symmetric with 1's in the corners, so that its eigenvalues are also 1 and -1. We are therefore interested in the $2x2$ matrix

$$\left[ \begin{array}{cc} 1 - p & p \\ p & p - 1 \end{array} \right]$$

Figure: Eigenvalue crossing for differently oriented eigenvectors

whose characteristic equation is

$$\left\vert \begin{array}{cc} 1 - p - \lambda & p \\ p & p - 1 - \lambda \end{array} \right\vert$$

or

$$\begin{eqnarray*} 2 (p - \frac{1}{2})^2 - \lambda^2 & = & \frac{1}{2}, \end{eqnarray*}$$

which is the equation of a hyperbola opening upwards and downwards, with vertex at $( 0, 1/2 )$ and approaching the x-axis no closer than $1/\surd 2$. The lines of the previous diagram are now asymptotes, but the interpolating lines no longer cross and the energy levels keep their places, so to speak. That is the content of the ``no crossing rule''no-crossing rule which specifies that this is a general proposition so the eigenvalues will retain their relative order even when the two matrices do not commute. Unless some common eigenvectors still remain, that is.