 # Lecture 7 Intersection of Hyperplanes and Matrix Inverse Shang-Hua Teng.

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Lecture 7 Intersection of Hyperplanes and Matrix Inverse Shang-Hua Teng

Elimination Methods for 2 by 2 Linear Systems 2 by 2 linear system can be solved by eliminating the first variable from the second equation by subtracting a proper multiple of the first equation and then by backward substitution Sometime, we need to switch the order of the first and the second equation Sometime we may not be able to complete the elimination

Singular Systems versus Non-Singular Systems A singular system has no solution or infinitely many solution –Row Picture: two line are parallel or the same –Column Picture: Two column vectors are co- linear A non-singular system has a unique solution –Row Picture: two non-parallel lines –Column Picture: two non-colinear column vectors

Gaussian Elimination in 3D Using the first pivot to eliminate x from the next two equations

Gaussian Elimination in 3D Using the second pivot to eliminate y from the third equation

Gaussian Elimination in 3D Using the second pivot to eliminate y from the third equation

Now We Have a Triangular System From the last equation, we have

Backward Substitution And substitute z to the first two equations

Backward Substitution We can solve y

Backward Substitution Substitute to the first equation

Backward Substitution We can solve the first equation

Backward Substitution We can solve the first equation

Generalization How to generalize to higher dimensions? What is the complexity of the algorithm? Answer: Express Elimination with Matrices

Step 1 Build Augmented Matrix Ax = b [A b]

Pivot 1: The elimination of column 1

Pivot 2: The elimination of column 2 Upper triangular matrix

Backward Substitution 1: from the last column to the first Upper triangular matrix

Expressing Elimination by Matrix Multiplication

Elementary or Elimination Matrix The elementary or elimination matrix That subtracts a multiple l of row j from row i can be obtained from the identity matrix I by adding (-l) in the i,j position

Elementary or Elimination Matrix

Pivot 1: The elimination of column 1 Elimination matrix

The Product of Elimination Matrices

Elimination by Matrix Multiplication

Linear Systems in Higher Dimensions

Booking with Elimination Matrices

Multiplying Elimination Matrices

Inverse Matrices In 1 dimension

Inverse Matrices In high dimensions

Inverse Matrices In 1 dimension In higher dimensions

Some Special Matrices and Their Inverses

Inverses in Two Dimensions Proof:

Uniqueness of Inverse Matrices

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 1D 2D

Gauss-Jordan Elimination for Computing A -1 3D

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

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