# Perturbing Diagonal Matrices

In the case of the example on page 51 (the 4x4 diagonal matrix with diagonal entries -1, -2, 0, -3), the equivalent strongly diagonal matrix can be found by entering A, then G = newhat(A), then one rcop, and then using the algorithm of Chapter 2, provided the rcop is well chosen. For instance, G = rcop(G,1,1,4,1) works, as you will find if you enter it followed by G = algo(G,100).

The idea is that the one rcop pushes the equilibrium (diagonal) matrix far enough away from equilibrium that the use of the algorithm of Chapter 2 not only restores it to equilibrium (diagonal) form but to stable equilibrium (strongly diagonal) form.

Try this experiment in other cases. That is, pick a diagonal matrix that is not strongly diagonal, and see if you can find one rcop that produces a matrix with the property that application of the algorithm of Chapter 2 to it produces a strongly diagonal matrix (and therefore produces the unique strongly diagonal matrix equivalent to the given diagonal matrix).

In terms of the metaphor of equilibrium, steps of type 4, 5, 6 only give a small perturbation and usually allow the algorithm to find another unstable---but slightly more stable---equilibrium, until finally the stable equilibrium solution is found.