Lecture 8 Matrix Inverse and LU Decomposition Shang-Hua Teng

Inverse Matrices In high dimensions

Uniqueness of Inverse Matrices

Inverse and Linear System

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

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

One More Property Proof So

Gauss-Jordan Elimination for Computing A-1

Gauss-Jordan Elimination for Computing A-1

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

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

Example:Gauss-Jordan Elimination for Computing A-1

Example:Gauss-Jordan Elimination for Computing A-1

Example:Gauss-Jordan Elimination for Computing A-1

Elimination = Factorization A = LU

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

What is the Inverse of Eij

Product with Elimination Matrices

Inverse of Triangular Matrix Call this matrix L

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

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

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)

Row Exchange and Permutation Matrix Define i We have:

Row Exchange and Permutation Matrix So:

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

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

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