## 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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