Conclusions

We conclude the paper with the following conjecture on the pseudo identity pair.

C onjectu re 1 If m — 2, for any pseudo identity matrix pair (C(z) ,D(z )), there must exist nilpotent matrices JV2, • • •, N q, N f = 0, i — 1,2, • • •, q, such that

C(z) = L Nq(z) ■ ■ ■ L N2(z)LNl(z), D(z) = R Nq(z) • • ■ R N2(z)RNl(z),

with

L Ni (z) = I 2 - N i + NiZ~ki, R Ni (z) = -N * z k> + I 2 + N*,

where ki E N, for i = 1,2, • • •, q, and

k\ + k2 + • • • + kq = kc = —ktf.