lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
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© 2012 Pearson Education, Inc.
Math 337-102Lecture 6
LU Factorization
Computer Graphics
Determinants
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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
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LU Example
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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.
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Finding L and U - Example
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Using LU to solve Ax = b
Factor A as LU Solve Ly = b for y Solve Ux = y for x
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Computer Graphics
Computer Graphics
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Translations – Homogeneous Coordinates
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Determinants
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Determinants – Co-Factors
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Determinants by Co-Factor Expansion
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Cofactor Expansion Example
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Cofactor Expansion Example
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Determinants of Triangular Matrices
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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|
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Row Operations and Determinants
Row replacement does not change determinant Row interchange negates the determinant Scaling – think of as factoring
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