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Page 1: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 2: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 3: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix

LU DecompositionSummary

Recall: If we want to solve Ax = b, we can:

• row reduce (A|b), or• find A−1.

Today: the method of LU decomposition.

Computational complexity of row reduction: n4/3Computational complexity of LU decomposition: 4n3/3

Page 4: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix

LU DecompositionOutline

• LU decompositions

• Using LU decompositions to solve Ax = b

• Finding LU decompositions: an example when A is square

• Finding LU decompositions: an example when A is not a square

• Application to electrical engineering (circuits)

• What do do when there are row swaps

Page 5: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 6: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 7: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 8: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 9: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 10: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 11: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 12: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 13: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix
Page 14: LU Decomposition - Peoplepeople.math.gatech.edu/.../notes/Feb22.pdf · LU Decomposition An LU factorization of A = m x n 's an expression where L x m unit lower triangular matrix

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