17.1 Problem statement

Let Hxx, Hxz, Hzx and Hzz be defined as follows.

Hxx  = E11 A + AT  E11 + CT1 C1 + ET12bC2 +  CT2 bT ET12
                    +E   B  bT ET  + E   B  BT E   +
                       11  1    12    11  1  1   11
                           E12 bbT ET12 + E12 bBT1 E11

                 T         T
Hxz  = E21 A +  a-(E21-+-E-12)+  cT C1 + E22 b C2 + cT BT ET +
                      2                                2   11
                       E21 B1 bT (E21 + ET12)          T   T
                       ----------2----------+  E21B1 B 1 E 11 +
                                   T          T
                             E22-bb--(E21-+-E12) + E22 bBT  ET
                                      2                   1  11

        T   T     T    (E12-+-ET21)a               T  T  T
Hzx = A   E21 + C 1 c +      2       + E11 B2 c + C2 b  E22 +
                              E   B  bT ET + E   B  BT  ET +
                                11  1     22    11  1  1  21
                  (E12-+-ET21)bbT-ET22    (E12-+--ET21)bB1T--ET21
                           2          +           2

Hzz = E22 a + aT ET22 + cT c + E21 B2 c + cT BT2 ET21 +
                            T  T            T  T
                    E21 B1 b E 22 + E21 B1 B 1 E21 +
                          E22 bbT ET22 + E22 bBT1 ET21

The math problem we address is:

(HGRAIL)    

Let A, B1, B2, C1, C2 be matrices of compatible size be given. Solve Hxx = 0, Hxz = 0, Hzx = 0, and Hzz = 0 for a, b, c and for E11, E12, E21 and E22. When can they be solved? If these equations can be solved, find formulas for the solution.