MATLAB: Matrices with unknown elements multiplication

Question

Excuse me i have a question

if i have a similar form of L-U decompostion which is LU=A such that L,U and A are all matrices while L and A are known

but unlike the usual case there is some known elements in the U matrix and i want to find the rest of unknowns in it

how do i do that in matlab please

Answer 1

Define the eements of the matrices in terms of symbolic unknowns. Then use solve to determine the unknown elements. Note that unless there are EXACTLY n^2 unknown elements remaining between the matrices L and U, your call to solve must fail.

Note: I can probably do this more efficiently, but...

L = tril(sym('L',[3,3]))
L = 
U = triu(sym('U',[3,3]))
U = 
U = U + diag([1 2 3]) - diag(diag(U))
U = 

So I have created unknown matrices L and U. I've arbitrarily set the diagonal elements of U to the numbers 1,2, and 3.

A = magic(3)
A = 3×3
     8     1     6
     3     5     7
     4     9     2

Now, can we solve for the remaining unknowns, such that L*U = A?

syms L1_1 L2_1 L3_1 L2_2 L3_2 L3_3 U1_2 U1_3 U2_3
LUparams = solve(L*U == A)
LUparams = struct with fields:
    L1_1: [1×1 sym]
    L2_1: [1×1 sym]
    L2_2: [1×1 sym]
    L3_1: [1×1 sym]
    L3_2: [1×1 sym]
    L3_3: [1×1 sym]
    U1_2: [1×1 sym]
    U1_3: [1×1 sym]
    U2_3: [1×1 sym]
L = subs(L,LUparams)
L = 
U = subs(U,LUparams)
U = 

Was I successful?

L*U - A
ans = 

Look at that. I got lucky.