Solving Matrices Equations: Finding X for A*X*B^T = C

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Homework Statement

1)Find all matrices X that satisfy the equation A*X*B^T = C, in terms of the LU
factorizations of A and B. State the precise conditions under which there are no
solutions.

B^T is the transpose of B.

2) Let U_1 and U_2 be two upper-triangular matrices. Let Z be an m × n matrix. Let
X be an unknown matrix that satisfies the equation
U_1X + XU_2 = Z.
A. Give an algorithm to find X in O(mn(m+ n)) flops (floating-point operations).
B. Find conditions on U_1 and U_2 which guarantee the existence of a unique solution
X.
C. Give a non-trivial example (U_1 is not equal to 0, U_2 is not equal to 0, X is not equal to 0) where those conditions are
not satisfied and
U_1X + XU_2 = 0.

Homework Equations

The Attempt at a Solution

any hints?

 

[oxxiissiixxo]

anyone has any hints? how should i attempt these problems?