Overview

The set of n linear equations represented by the single matrix equation

where A is a square n n matrix and ? is a constant, permits of solutions other than the trivial solution x = 0 if and only if

where I is the unit matrix. This is the characteristic equation of the matrix A. If we expand the determinant by Cramer's rule ( cf chapter 10) we see that p _A is a polynomial of degree n in ?. The characteristic equation therefore possesses n roots ? _i, not all necessarily distinct from each other, which are the eigenvalues of the matrix A. Correspondingly, there are n eigenvectors x ^{( i)}, not necessarily all distinct, which satisfy the relations

Note that the eigenvectors are undetermined to a multiplicative factor. This may be removed by a consistent choice of normalization; and it is customary to set ? x ^{( i)} ? _? = 1 or ? x ^{( i)} ? ₂ = 1 depending on the ease of computation and the use to which the results are to be put.

A set of vectors x ^{( k)} is said to be linearly independent if the equation ? _k c _k x ^{( k)} = 0 has no solution other than c = 0; otherwise they are...