Eigencomponents

The fth diagonal element At of the matrix A is called the ith eigenvalue of A. The column-vectors ( . ) of U are usually taken (although not necessarily) of unit length and are called the eigenvectors of the matrix A. 

A matrix and its inverse therefore have the same eigenvectors but reciprocal eigenvalues. Each pair of eigenvector and eigenvalue (eigencomponents) can be related through a linear 

The order of eigencomponents is arbitrary, although they are commonly ranked in the order of increasing or decreasing eigenvalues. Some properties result from those of the determinant, e.g., 

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If A is a symmetric matrix, then the matrix V is orthogonal. This can be shown by considering two eigencomponent pairs i and j... 

Given the eigencomponent decomposition of the symmetric covariance matrix... 

An eigencomponent routine confirms that the matrix ATA has one eigenvalue equal to zero with corresponding eigenvector [0.485, —0.485, +0.2425, —0.485, -0.485]1. It is common practice to use integers as stoichiometric coefficients. This can be achieved by dividing each component by the component of smallest modulus (0.2425), which produces the vector [2, - 2,1, - 2, - 2] corresponding to the mineral reaction... 

In multicomponent systems, the single diffusivity is replaced by a multicomponent diffusion matrix. By going through similar steps, it can be shown that the [D] matrix must have positive eigenvalues if the phase is stable. In a multicomponent system, the diffusive flux of a component can be up against its chemical potential gradient except for eigencomponents. 

The correspondence between a particular diagonal element and the associated column and row is not affected, and so we are free to take the triples of eigenvalue, column and row in whatever sequence we like. In this book we call such a triple an eigencomponent. 

At every multiplication the weights get multiplied by the eigenvalues. Thus as the number of multiplications increases, the contribution of the eigenvector with the largest eigenvalue gets to be more and more dominant. This is why that eigencomponent is called the dominant one. 

If, however, the original vector happened to be orthogonal to the dominant eigenrow, then there would be nothing of that component to grow relative to the others, and in those circumstances the second eigencomponent will dominate. 

To explore further, we choose our coordinate system so that this straight line is the. x-axis. The subdominant eigencomponent then makes no contribution to y, which is dominated by the third eigencomponent. The column eigenvector looks complicated, but in fact it is just a quadratic variation, with an offset added... 

This procedure can be applied to any primal binary scheme, although it may be necessary to imagine higher dimensions than 3 in order to keep applying the principle of suppressing successive dominant eigencomponents. 

Why do all the arithmetic of calculating eigencomponents, when there is software available to do it for us ... 

The reason for this tedious working through is that the separations we have made by symmetry and by block structure do enable us to pick out and observe patterns in the eigenvectors which could be confused when there are two or more eigencomponents with the same eigenvalue. These patterns are going to be significant in a couple of chapters time, and it is important to observe them empirically first. 

In some rather special cases, it is possible for the matrices on some diagonals to have only polynomial eigencomponents, saying that the Holder continuity is infinite. This happens for the B-splines. 

Note that the unit eigencolumn vanishes in a puff of smoke, because its first differences are all zero. Yes, a column of zeroes is an eigenvector, but it is the trivial one, not to be considered beside the real ones. The number of eigencomponents of the divided difference scheme is therefore one less than the number in the original scheme. 

The eigenrows are somewhat easier in that we have to antidifference on the way up the chain and therefore only have to difference on the way back down. The complication takes a different form, deciding what eigenrow is going to apply to each unit eigencomponent as it gets inserted. It turns out... 

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