A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). 4 3 (a) Verify that A = -4 -5 7 Is a product of a L=-1 -G 8 6 -8 100] [4 3 -5 10 and U = 0 -2 2 201 0 0 2 (b) Use the LU factorization to solve Az = H -4 for . To do so, first solve Ly = 6, and after Uz = y. (c) Find L-¹ and U-¹, then use these to find A-¹ showing that A is invertible.
A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). 4 3 (a) Verify that A = -4 -5 7 Is a product of a L=-1 -G 8 6 -8 100] [4 3 -5 10 and U = 0 -2 2 201 0 0 2 (b) Use the LU factorization to solve Az = H -4 for . To do so, first solve Ly = 6, and after Uz = y. (c) Find L-¹ and U-¹, then use these to find A-¹ showing that A is invertible.
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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Transcribed Image Text:4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above
diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal).
(a) Verify that A
4
-4
8
-5
-5 7 Is a product of a L
=
43-
6
(b) Use the LU factorization to solve Az =
0 0
-1 1 0 and U =
2 0 1
4
3
0 -2 2
0 0 2
for . To do so, first solve Ly=b, and after Ur = y.
-1
(c) Find L-¹ and U-1, then use these to find A-¹ showing that A is invertible.
(d) (Optional) To find the LU factorization, reduce A to U by using only row replacements operations. Second,
place entries in L so that the same sequence of row operations reduces L to I. Follow this algorithm to find the
LU decomposition of A. (Consult Section 2.5 for more details.)
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