Eigenvector Decomposition and Canonical Basis Representation in Matrix Analysis

The eigenvector pν can be expressed as pν = (pA,ν , pB,ν )T, where pA,ν is the subvector corresponding to the first M coordinates and pB,ν is the subvector corresponding to the remaining N-M coordinates.

Furthermore, we can write pν as a linear combination of the canonical basis vectors as follows:

pν = ∑k=1M pkν ek + ∑k=M+1N pkν ek

where pkν is the kth coordinate of pν.

Note that since the νth coordinate of pν is nonnegative, we have pkν ≥ 0 for all k.

Also, since pν is an eigenvector corresponding to the eigenvalue λν, we have:

ApA,ν = λνpA,ν BpB,ν = λνpB,ν ApB,ν = 0

where ApA,ν and BpB,ν are the submatrices of A and B corresponding to pA,ν and pB,ν, respectively, and ApB,ν is the cross-product matrix.

Using these equations and the fact that pν can be expressed as a linear combination of the canonical basis vectors, we can derive the following system of equations for the coefficients pkν:

Apkν = λνpkν for k=1,2,...,M Bpkν = λνpkν for k=M+1,M+2,...,N Apkν = 0 for k=1,2,...,M and l=M+1,M+2,...,N

Solving this system of equations for pkν yields the coefficients of the linear combination of canonical basis vectors that make up pν.