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4.4-5 Prove that a triangular factorization is unique in the following sense: If A is invertible and L,U¡ = A = L,U, where L, L, are unit-lower-triangular matrices and U, U, are upper-triangular matrices, then L, hat U, L, 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, L'L, - I.) L2 and U, U. (Hint: Use Exercise 4.1-8 to prove

4.4-5 Prove that a triangular factorization is unique in the following sense: If A is invertible and L,U¡ = A = L,U, where L, L, are unit-lower-triangular matrices and U, U, are upper-triangular matrices, then L, hat U, L, 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, L'L, - I.) L2 and U, U. (Hint: Use Exercise 4.1-8 to prove

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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,
Transcribed Image 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,
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