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

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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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