Why does lu function yield different lower triangle matrix if I return [L,U] rather than [L, U, P]?

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% square matrix A
A=[10,-7,0;-3,2,6;5,-1,5]
A = 3×3
10 -7 0 -3 2 6 5 -1 5
Return only L and U
[L1,U1] = lu(A);
Return L, U and P
[L2,U2,P2] = lu(A);
Compare L1 and L2
L1
L1 = 3×3
1.0000 0 0 -0.3000 -0.0400 1.0000 0.5000 1.0000 0
L2
L2 = 3×3
1.0000 0 0 0.5000 1.0000 0 -0.3000 -0.0400 1.0000

Accepted Answer

Christine Tobler
Christine Tobler on 18 May 2022
The LU decomposition really involves three new matrices: An upper-triangular matrix U, a lower-triangular matrix L, and a permutation matrix P. This is what the three-output syntax returns:
A = [10,-7,0;-3,2,6;5,-1,5]
A = 3×3
10 -7 0 -3 2 6 5 -1 5
[L, U, P] = lu(A)
L = 3×3
1.0000 0 0 0.5000 1.0000 0 -0.3000 -0.0400 1.0000
U = 3×3
10.0000 -7.0000 0 0 2.5000 5.0000 0 0 6.2000
P = 3×3
1 0 0 0 0 1 0 1 0
P*A
ans = 3×3
10 -7 0 5 -1 5 -3 2 6
L*U
ans = 3×3
10.0000 -7.0000 0 5.0000 -1.0000 5.0000 -3.0000 2.0000 6.0000
Unfortunately, lu also has a 2-output syntax. However, since it wouldn't be numerically safe to just compute L and U without a permutation matrix, internally we still compute all three matrices, and then return the first output as P'*L
[L2, U2] = lu(A);
L2
L2 = 3×3
1.0000 0 0 -0.3000 -0.0400 1.0000 0.5000 1.0000 0
P'*L
ans = 3×3
1.0000 0 0 -0.3000 -0.0400 1.0000 0.5000 1.0000 0
L2*U
ans = 3×3
10.0000 -7.0000 0 -3.0000 2.0000 6.0000 5.0000 -1.0000 5.0000
A = 3×3
10 -7 0 -3 2 6 5 -1 5
So the result of two-output LU satisfies A == L*U, but the output L isn't a lower-triangular matrix as one might expect.
You can argue that it would be better if the LU function didn't have a two-output syntax at all, but that decision was made a long time ago, and was probably based on the point that most people think of LU as a two-matrix decomposition, without thinking of the necessary permutation vector.

More Answers (1)

Steven Lord
Steven Lord on 18 May 2022
From the documentation page for the lu function:
"[L,U] = lu(A) returns an upper triangular matrix U and a matrix L, such that A = L*U. Here, L is a product of the inverse of the permutation matrix and a lower triangular matrix.
[L,U,P] = lu(A) returns an upper triangular matrix U, a lower triangular matrix L, and a permutation matrix P, such that P*A = L*U. The syntax lu(A,'matrix') is identical."
Emphasis added.

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