Let A be an invertible matrix such that , where L and L1 are unit lower triangular matrices, and U and U1 are upper triangular matrices. Show that L = L1 and U = U1.
Solution
A = L1U1 = L2U2 are two LU-factorizations of the nonsingular matrix
A.The equation L1U1 = L2U2 can be written in the form L12 L1 = U2U11
, where
by lemmas 1.2-1.4 L1
2 L1 is unit lower triangular and U1
2 U1 is upper triangular.
But then both matrices must be diagonal with ones on the diagonal.
We conclude that L12 L1 = I = U1U21
which means that L1 = L2 and
U1 = U2.
.
Let A be an invertible matrix such that - where L and L1 are unit low.docx
1. Let A be an invertible matrix such that , where L and L1 are unit lower triangular matrices, and
U and U1 are upper triangular matrices. Show that L = L1 and U = U1.
Solution
A = L1U1 = L2U2 are two LU-factorizations of the nonsingular matrix
A.The equation L1U1 = L2U2 can be written in the form L12 L1 = U2U11
, where
by lemmas 1.2-1.4 L1
2 L1 is unit lower triangular and U1
2 U1 is upper triangular.
But then both matrices must be diagonal with ones on the diagonal.
We conclude that L12 L1 = I = U1U21
which means that L1 = L2 and
U1 = U2.
