1. Lecture 8 Matrix Inverse and LU Decomposition
Shang-Hua Teng

2. Inverse Matrices
In high dimensions

3. Uniqueness of Inverse Matrices

4. Inverse and Linear System

5. Inverse and Linear System
Therefore, the inverse of A exists if and only if elimination produces n non-zero pivots (row exchanges allowed)

6. Inverse, Singular Matrix and Degeneracy
Suppose there is a nonzero vector x such that Ax = 0 [column vectors of A co-linear] then A cannot have an inverse
Contradiction:
So if A is invertible, then Ax =0 can only have the zero solution x=0

7. One More Property
Proof So

8. Gauss-Jordan Elimination for Computing A-1

9. Gauss-Jordan Elimination for Computing A-1

10. Gauss-Jordan Elimination for Computing A-1
3D: Solving three linear equations defined by A simultaneously
n dimensions: Solving n linear equations defined by A simultaneously

11. Example:Gauss-Jordan Elimination for Computing A-1
Make a Big Augmented Matrix

12. Example:Gauss-Jordan Elimination for Computing A-1

13. Example:Gauss-Jordan Elimination for Computing A-1

14. Example:Gauss-Jordan Elimination for Computing A-1

15. Elimination = Factorization A = LU

16. Elimination = Factorization A = LU
What is the inverse of the left triangle?
Call the upper triangular matrix U

17. What is the Inverse of Eij

18. Product with Elimination Matrices

19. Inverse of Triangular Matrix
Call this matrix L

20. Matrix L
L is a lower triangular matrix with 1’s on the diagonal and the non-zeros entries are exactly the multipliers lij

21. If the elimination needs no row exchange
LU Factorization
And ….
A = LU
If the elimination needs no row exchange

22. Two Step Method
1. Factor A = LU by elimination possible with row permutation
2. Solve Lc = b (by forward substitution), then Ux = c (by backward substitution).
Complexity: step 1 (multiplications & subtractions)
Step 2: O(n2)

23. Row Exchange and Permutation Matrix
Define i
We have:

24. Row Exchange and Permutation Matrix
So:

25. Row Exchange and Permutation Matrix
To Permute, we just need to assemble a matrix of ei properly
For example, to move row 1 to row 3, row 2 to row 1, and
row 3 to row 2, we can use

26. Need to Permute Rows In Elimination
0 pivot
Permute row 2 with row 3 we have

27. LU Factorization Theorem
For any matrix A, there exists a permutation P, such that
PA = LU
Can be obtained by elimination with potential row exchange