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Numerical Computation

Numerical Computation. Lecture 7: Finding Inverses: Gauss-Jordan United International College. Review. During our Last Two Classes we covered: Linear Systems : LU Factorization (or Decomposition). Today. We will cover: Gauss-Jordan Method for finding Inverses. Review.

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Numerical Computation

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  1. Numerical Computation Lecture 7: Finding Inverses: Gauss-Jordan United International College

  2. Review • During our Last Two Classes we covered: • Linear Systems: LU Factorization (or Decomposition)

  3. Today • We will cover: • Gauss-Jordan Method for finding Inverses

  4. Review

  5. Row Operations as Matrices Definition: An n×n matrix is called an elementary matrixif it can be obtained from the n×n identity matrix I by performing a single elementary row operation.

  6. Row Operations as Matrices Row Operations: Interchange two rows Add r * times one row to another row Multiply one row by a scalar

  7. Row Operations as Matrices If the elementary row operation matrix E results from performing a certain row operation on the identity I, and if A is an m×n matrix ,then the product EA is the matrix that results when this same row operation is performed on A . That is, when a matrix A is multiplied on the left by an elementary matrix E ,the effect is to perform an elementary row operation on A .

  8. Row Operations as Matrices Example: For the matrix Consider the elementary matrix This matrix is obtained from I by adding 3*row 1 to row 3. Note that EA is the matrix that results from adding 3*row 1 to row 3.

  9. Gauss-Jordan Method To find the inverse to an nxn matrix A: Adjoin the identity matrix I to the right side of A, thereby producing a matrix of the form Apply row operations to this matrix until the left side is reduced to I. If successful, these operations will convert the right side to A-1 ,so that the final matrix will have the form

  10. Gauss-Jordan Method Example:

  11. Gauss-Jordan Method Example: Do row operations to get upper triangular form: (Like Gaussian Elimination)

  12. Gauss-Jordan Method Example: Continue doing row operations to get 0’s in columns above the pivots:

  13. Gauss-Jordan Method Example: At this point the last matrix on the left is the Identity. Thus, the right matrix must be the inverse to A:

  14. Gauss-Jordan Method Example:

  15. Matlab Implementation • Task: Implement Gaussian Jordan method in a Matlab M-file. • Notes • Input = Coefficient matrix A • Output = Inverse Matrix A-1 • Discussion: How can we modify our Gaussian Elimination code to do this?

  16. Why does Gauss-Jordan Work? Recall: Adjoin the identity matrix I to the right side of A, thereby producing a matrix of the form Apply row operations to this matrix until the left side is reduced to I.

  17. Why does Gauss-Jordan Work? Recall: Every row operation applied to [A | I] can be represented by an elementary matrix E. That is, the row operation is equivalent to E* [A | I]. Thus, Gauss-Jordan can be viewed as a series of matrix operations Ep Ep-1 … E1 [A | I] = [I | B] But, this means that Ep Ep-1 … E1 A = I. Then, Ep Ep-1 … E1 = A-1 . Thus, the right side of [A | I] is transformed to B = Ep Ep-1 … E1 I = A-1

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