4.4-5 Prove that a triangular factorization is unique in the following sense: If A is invertible and L,U1 = A = L,U, where L1, L2 are unit-lower-triangular matrices and U, U, are U. (Hint: Use...

Extracted text: 4.4-5 Prove that a triangular factorization is unique in the following sense: If A is invertible and L,U1 = A = L,U, where L1, L2 are unit-lower-triangular matrices and U, U, are U. (Hint: Use Exercise 4.1-8 to prove upper-triangular matrices, then L, that U, L2 must be invertible; then show that L'L, - U,U,' must hold, which implies, with Exercise 4.4-4, that L,'L,¸ must be a diagonal matrix; hence, since both L, and L, have l's on their diagonal, L7 'L, - 1.) L, and U,