Solving a linear system equations with variables on both sides

I am trying to solve a linear system of equation in which variables occur on both sides.

Lu = [Lu1; Lu2; Lu3]
A = [1 2 3; 4 5 6; 7 8 9];
B = [U1; U2; U3];

In this system, I know Lu1, U2, and U3 and none of them is zero. Is there a way to solve this system of equations?

Answer 1

[Walter Roberson]

If the system is Lu'*A == B, then that is 9 equations in 3 unknowns. If you proceed to solve one variable at a time, then after you have solved for all three variables you reach the system

[ U2 == U2, U2 == U2, U2 == U2]
[ U2 == U2, U2 == U2, U2 == U2]
[ U2 == U3, U2 == U3, U2 == U3]

so the system can only be solved in the case that U2 == U3

[joe kangas]

this is a system with three unknowns and three equations, not 9. This problem comes up a lot in finite element solutions and as far as I'm aware there's not a straight forward general soltution. I believe they are itterative

[Walter Roberson]

If the system is Lu'*A == B' then

Lu = sym('Lu', [3 1], 'real');
A = [1 2 3; 4 5 6; 7 8 10];       %note original 7 8 9 is not full rank
B = sym('U', [3,1], 'real');
Lu'*A == B'

ans = 

So that is three equations in six unknowns.

%Lu' == B' * inv(A)
%Lu' == B'/A
%Lu = (B'/A)'
left_LU_solution = (B'/A)'

left_LU_solution = 

On the other hand if the equations were

A*Lu == B

ans = 

Then that would be a system of 9 equations in 6 unknowns:

right_Lu_solultion = A\B

right_Lu_solultion = 

Answer 2

[Bruno Luong]

See my answer in another thread.

Answer 3

[Paul]

Similar problem discussed here