Linear Alegbra and Its Applications, Exercise 1.6.17

Exercise 1.6.17. (a) Suppose that the n by n matrix A can be factored as LDU, where L and U have ones on the diagonal. Factor the transpose of A.

(b) If we have

A^Ty = b

what triangular systems will provide a solution?

Answer: (a) Since A can be factored as LDU we have

A^T = (LDU)^T = U^T(LD)^T = U^T(D^TL^T) = U^TDL^T

(Note that D is its own transpose.)

Since U is upper triangular its transpose is lower triangular, and since L is lower triangular its transpose is upper triangular. We thus have the factorization

A^T = U^TDL^T

(b) Since A^T can be factored as shown above, we have

A^Ty = b \rightarrow U^TDL^Ty = b \rightarrow U^Tc = b \ \rm and \ DL^Ty = c

These are the two triangular systems as requested.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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