TABLE OF CONTENTS When eigenvalues of the matrix A are repeated with a multiplicity of r, some of the eigenvectors may be linearly dependent on others. Guidance as to the number of linearly independent eigenvectors can be obtained from the rank of the matrix A.
As shown in Sections 5.6 and 5.8, a set of simultaneous, linear homogeneous algebraic equations, if consistent, produces a unique solution if the rank of the n n coefficient matrix is equal to its order. If the rank of the coefficient matrix is less than its order, an infinite number of solutions is produced.
To determine how many linearly independent eigenvectors are associated with each repeated eigenvalue, it is necessary to examine the rank of the matrix K = [ A ? ? I]. The first step is to form K with the repeated eigenvalue inserted. Then, the rank of K is determined and it is found that the number of linearly independent eigenvectors associated with the repeated eigenvalue will be equal to the difference between the order of K and the rank of A, that is, n ? r.
For the matrix
the characteristic matrix is
The characteristic equation is obtained by setting the determinant of the characteristic matrix equal to zero.
and this yields three eigenvalues, one of which is repeated
With ? 1 = ? 2 = 1,
and for this particular K
