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lec19.pdf - MATH 3607 Lecture 19 Tae Eun Kim Monday Tae Eun...

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MATH 3607 Lecture 19 Tae Eun Kim Monday, October 16, 2017 Tae Eun Kim (Ohio State) Lecture 19 Monday, October 16, 2017 1 / 24
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Linear algebra in MatLab Some common matrix functions in MatLab includes inv, det, eig , etc. Examples Define a random matrix A = rand(4) and try the following statements: >> Ai = inv(A) % inverse of A >> A * Ai - eye(size(A)) % zero for nonsingular A >> det(A) % determinant of A >> eig(A) % eigenvalue of A Note: Recall that a square matrix A is singular if any one of the following equivalent conditions is met: det( A ) = 0 . Ax = 0 has a nonzero solution. A has a zero eigenvalue. Tae Eun Kim (Ohio State) Lecture 19 Monday, October 16, 2017 2 / 24
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Solutions to linear system of equations Consider the system of linear equations represented as a matrix equation A x = b , where A is the coe ffi cient matrix, b is the right-hand sides of the equations, and x is the vector of unknowns. The system either has a unique solution x if A is invertible or nonsingular, in which case x = A 1 b , or has no solution or infinitely many solutions x if A is not invertible or singular. The Backslash operator " \ " In order to solve for x in MatLab , we use the backslash symbol “ \ ”: >> x = A \ b This produces the solution using Gaussian elimination, without explicitly forming the inverse of A . Warning: Even though x = A 1 b analytically, don’t use x = inv(A) * b ! Tae Eun Kim (Ohio State) Lecture 19 Monday, October 16, 2017 3 / 24
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Gaussian elimination Gaussian elimination is an algorithm for solving a system of linear equations that involves a sequence of row operations perfomed on the associated matrix of coe ffi cients. This is also known as the method of row reduction. There are three variations to this method: 1 G.E. without pivoting 2 G.E. with partial pivoting 3 G.E. with full pivoting Tae Eun Kim (Ohio State) Lecture 19 Monday, October 16, 2017 4 / 24
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1. Gaussian elimination without pivoting Example: Solve the following system of equations. 2 x 1 + 2 x 2 + x 3 = 6 4 x 1 + 6 x 2 + x 3 = 8 5 x 1 5 x 2 + 3 x 3 = 4 matrix equation ≠≠≠≠≠≠≠≠≠≠æ Q a 2 2 1 4 6 1 5 5 3 R b ¸ ˚˙ ˝ A Q a x 1 x 2 x 3 R b ¸ ˚˙ ˝ x = Q a 6 8 4 R b ¸ ˚˙ ˝ b Step 1: Write the equation in the augmented matrix form and reduce it to an echelon form via row operations. A 2 2 1 6 4 6 1 8 5 5 3 4 B ≠æ A 2 2 1 6 0 10 3 4 0 10 0 . 5 1 B ≠æ A 2 2 1 6 0 10 3 4 0 0 3 . 5 7 B . Step 2: From the reduced augmented matrix, we can solve for x 3 , then x 2 , and then x 1 . This processes is called backward substitution . x = (3 , 1 , 2) T . Tae Eun Kim (Ohio State) Lecture 19 Monday, October 16, 2017 5 / 24
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As shown in the example, a general Gaussian elimination (without pivoting) involves two steps: 1 Row reduction: transform A x = b ≠æ U x =  b where U = Q c c c a u 11 u 12 · · · u 1 n u 22 u 2 n . . . . . . 0 u nn R d d d b and  b = Q c c c a  b 1  b 2 .
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