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We study the joint low-rank factorization of the matrices X=[A B]G and Y=[A C]H, in which the columns of the shared factor matrix A correspond to vectorized rank-one matrices, the unshared factors B and C have full column rank, and the matrices G and H have full row rank. The objective is to find the shared factor A, given only X and Y. We first explain that if the matrix [A B C] has full column rank, then a basis for the column space of the shared factor matrix A can be obtained from the null space of the matrix [X Y]. This in turn implies that the problem of finding the shared factor matrix A boils down to a basic Canonical Polyadic Decomposition (CPD) problem that in many cases can directly be solved by means of an eigenvalue decomposition. Next, we explain that by taking the rank-one constraint of the columns of the shared factor matrix A into account when computing the null space of the matrix [X Y], more relaxed identifiability conditions can be obtained that do not require that [A B C] has full column rank. The benefit of the unconstrained null space approach is that it leads to simple algorithms while the benefit of the rank-one constrained null space approach is that it leads to relaxed identifiability conditions. Finally, a joint unbalanced orthogonal Procrustes and CPD fitting approach for computing the shared factor matrix A from noisy observation matrices X and Y will briefly be discussed.