In this problem, we prove that the LU decomposition of an invertible n n matrix is

Chapter 2, Problem 30

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In this problem, we prove that the LU decomposition of an invertible n n matrix is unique in the sense that, if A = L1U1 and A = L2U2, where L1, L2 are unit lower triangular matrices and U1, U2 are upper triangular matrices, then L1 = L2 and U1 = U2.(a) Apply Corollary 2.6.13 to conclude that L2 andU1 are invertible, and then use the fact thatL1U1 = L2U2 to establish that L12 L1 =U2U11 .(b) Use the result from (a) together with Theorem2.2.24 and Corollary 2.2.25 to prove thatL12 L1 = In and U2U11 = In, from which therequired result follows.