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

35 次查看（过去 30 天）
John Taranto
John Taranto 2022-5-18
评论： John Taranto 2022-5-18
Ran in:
% 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

请先登录，再回答此问题。

采纳的回答

Christine Tobler
Christine Tobler 2022-5-18
Ran in:
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.

更多回答（1 个）

Steven Lord
Steven Lord 2022-5-18
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.

类别

在 Help Center 和 File Exchange 中查找有关 Creating and Concatenating Matrices 的更多信息

标签

产品


版本

R2021a

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by