Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2

© 2012 Pearson Education, Inc.
Math 337-102
Lecture 6
LU Factorization
Computer Graphics
Determinants

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)

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.

LU Example

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.

Finding L and U - Example

Using LU to solve Ax = b
 Factor A as LU
 Solve Ly = b for y
 Solve Ux = y for x

Computer Graphics

Translations – Homogeneous Coordinates

Determinants

Determinants – Co-Factors

Determinants by Co-Factor Expansion

Cofactor Expansion Example

Determinants of Triangular Matrices

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|

Row Operations and Determinants
 Row replacement does not change determinant
 Row interchange negates the determinant
 Scaling – think of as factoring

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