Continuing Problem 5, we can extract some of the eigenvectors and eigenvalues of a kinship matrix of a general pedigree [16]. Consider a set of individuals in the pedigree possessing the same inbreeding coefficient and the same kinship coefficients with other pedigree members.

Typical cases are a set of siblings with no children and a married pair of pedigree founders with shared offspring but no unshared offspring.
Without loss of generality, number the members of the set 1,...,m and the remaining pedigree members m + 1,...,n. Show that

(a) The kinship matrix Φ can be written as the partitioned matrix Φ =  a1Im + a211t 1bt b1t C 
, where 1 is a column vector consisting of m 1’s, Im is the m × m identity matrix, a1 and a2 are real constants, b is a column vector with n−m entries, and C is the (n−m)×(n−m) kinship matrix of the n − m pedigree members not in the designated set.

(b) The matrix a1Im + a211t has 1 as eigenvector with eigenvalue a1 + ma2 and m − 1 orthogonal eigenvectors ui = 1 i − 1
i−1 j=1 ej − ei, i = 2,...,m, with eigenvalue a1. Note that each ui is perpendicular to 1.

(c) The m−1 partitioned vectors  ui 0 
are orthogonal eigenvectors of Φ with eigenvalue a1.