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 1 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=1' data-page='1' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/1.jpg' width='140' height='200' /> Page 2 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=2' data-page='2' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/2.jpg' width='140' height='200' /> Page 3 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=3' data-page='3' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/3.jpg' width='140' height='200' /> 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 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=4' data-page='4' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/4.jpg' width='140' height='200' /> 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 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=5' data-page='5' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/5.jpg' width='140' height='200' /> Page 6 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=6' data-page='6' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/6.jpg' width='140' height='200' /> Page 7 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=7' data-page='7' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/7.jpg' width='140' height='200' /> Page 8 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=8' data-page='8' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/8.jpg' width='140' height='200' /> Page 9 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=9' data-page='9' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/9.jpg' width='140' height='200' /> Page 10 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=10' data-page='10' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/10.jpg' width='140' height='200' /> Page 11 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=11' data-page='11' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/11.jpg' width='140' height='200' /> Page 12 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=12' data-page='12' class='cut-thumb lazyload' alt='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' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader042.cupdf.com/reader042/viewer/2022040215/5edca4e4ad6a402d66676622/html5/thumbnails/12.jpg' width='140' height='200' /> Page 13 <img data-url='document/lu-decomposition-lu-decomposition-an-lu-factorization-of-a-m-x-n-s-an-expression.html?page=13' data-page='13' class='cut-thumb lazyload' alt='Page 13: LU Decomposition - 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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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