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Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
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Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
1.
© 2012 Pearson
Education, Inc. Math 337-102 Lecture 6 LU Factorization Computer Graphics Determinants
2.
Slide 2.2- 2©
2012 Pearson Education, Inc. LU Factorization Factor A = LU A is mxn A does NOT have to be square!!! L is mxm (square) – Lower Triangular with 1’s on the diagonal U is mxn REF(A)
3.
Using LU to
Solve Ax = b Ax = b (LU)x = b L(Ux) = b Let y = Ux Solve Ly = b Then Ux = y Slide 2.2- 3© 2012 Pearson Education, Inc.
4.
LU Example Slide 2.2-
4© 2012 Pearson Education, Inc.
5.
Finding L and
U U is Row Echelon Form of A using row replacement only No interchanges or scaling!!!! L entries are such that the same sequence of row operations that reduce A to U will reduce L to I. Slide 2.2- 5© 2012 Pearson Education, Inc.
6.
Finding L and
U - Example Slide 2.2- 6© 2012 Pearson Education, Inc.
7.
Using LU to
solve Ax = b Factor A as LU Solve Ly = b for y Solve Ux = y for x Slide 2.2- 7© 2012 Pearson Education, Inc.
8.
Computer Graphics
9.
Computer Graphics Slide 2.2-
9© 2012 Pearson Education, Inc.
10.
Translations – Homogeneous
Coordinates Slide 2.2- 10© 2012 Pearson Education, Inc.
11.
Determinants Slide 2.2- 11©
2012 Pearson Education, Inc.
12.
Determinants – Co-Factors Slide
2.2- 12© 2012 Pearson Education, Inc.
13.
Determinants by Co-Factor
Expansion Slide 2.2- 13© 2012 Pearson Education, Inc.
14.
Cofactor Expansion Example Slide
2.2- 14© 2012 Pearson Education, Inc.
15.
Cofactor Expansion Example Slide
2.2- 15© 2012 Pearson Education, Inc.
16.
Determinants of Triangular
Matrices Slide 2.2- 16© 2012 Pearson Education, Inc.
17.
Row Operations and
Determinants Thm 3-3: Let A be a square matrix. a)If a multiple of one row is added to another row to produce a matrix B, then |B| = |A| b)If two rows are interchanged to produce B, then |B| = -|A| c)If one row of A is multiplied by k to produce B, then |B| = k|A| Slide 2.2- 17© 2012 Pearson Education, Inc.
18.
Row Operations and
Determinants Row replacement does not change determinant Row interchange negates the determinant Scaling – think of as factoring Slide 2.2- 18© 2012 Pearson Education, Inc.
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