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Lecture 6 Matrix Operations and Gaussian Elimination for Solving Linear Systems Shang-Hua Teng

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Matrix (Uniform Representation for Any Dimension) An m by n matrix is a rectangular table of mn numbers

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Matrix (Uniform Representation for Any Dimension) Can be viewed as m row vectors in n dimensions

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Matrix (Uniform Representation for Any Dimension) Or can be viewed as n column vectors in m dimensions

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Squared Matrix An n by n matrix is a squared table of n 2 numbers

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Some Special Squared Matrices All zeros matrix Identity matrix

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Matrix Operations Addition Scalar multiplication Multiplication

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1. Matrix Addition: Matrices have to have the same dimensions What is the complexity?

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2. Scalar Multiplication: What is the complexity?

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3. Matrix Multiplication Two matrices have to be conformal What is the complexity?

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Matrix Multiplication Two matrices have to be conformal

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The Laws of Matrix Operations A + B = B + A (commutative) c(A+B) = cA + c+B (distributive) A + (B + C) = (A + B) + C (associative) C(A+B) = CA + CB (distributive from left) (A+B)C = AC+BC (distributive from right) A(BC) = (AB)C (associative) But in general:

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Counter Example

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Special Matrices Identity matrix I –IA = AI = A Square Matrix A

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Elimination: Method for Solving Linear Systems Linear Systems == System of Linear Equations Elimination: –Multiply the LHS and RHS of an equation by a nonzero constant results the same equations –Adding the LHSs and RHSs of two equations does not change the solution

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Elimination in 2D Multiply the first equation by 3 and subtracts from the second equation (to eliminate x) The two systems have the same solution The second system is easy to solve

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Geometry of Elimination (3,1) 8y = 8 Reduce to a 1-dimensional problem.

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Upper Triangular Systems and Back Substitution Back substitution –From the second equation y = 1 –Substitute the value of y to the first equation to obtain x-2=1 –Solve it we have: x = 3 So the solution is (3,1)

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How Much to Multiply before Subtracting Pivot: first nonzero in the row that does the elimination Multiplier: (entry to eliminate) divided by (pivot) Multiply: = 3/1

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How Much to Multiply before Subtracting Pivot: first nonzero in the row that does the elimination Multiplier: (entry to eliminate) divided by (pivot) Multiply: = 3/2 The pivots are on the diagonal of the triangle after the elimination

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Breakdown of Elimination What is the pivot is zero == one can’t divide by zero!!!! Eliminate x: No Solution!!!!: this system has no second pivot

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Geometric Intuition (Row Pictures) Two parallel lines never intersect (3,1) 8y = 8

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Geometric Intuition (Column Picture) Two column vectors are co-linear!!!!

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Geometric Intuition Geometric degeneracy cause failure in elimination!

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Failure in Elimination May Indicate Infinitely Many Solutions y is free, can be number! Geometric Intuition (row picture): The two line are the same Geometric Intuition (column picture): all three column vectors are co-linear

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Failure in Elimination (Temporary and can be Fixed) First pivot position contains zero Exchange with the second equation Can be solved by backward substitution!

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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

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

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

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

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

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Backward Substitution And substitute z to the first two equations

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Backward Substitution We can solve y

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Backward Substitution Substitute to the first equation

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Backward Substitution We can solve the first equation

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Backward Substitution We can solve the first equation

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

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

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Pivot 1: The elimination of column 1

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Pivot 2: The elimination of column 2 Upper triangular matrix

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

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Expressing Elimination by Matrix Multiplication

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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 entry by adding (-l) in the i,j position

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Elementary or Elimination Matrix

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Pivot 1: The elimination of column 1 Elimination matrix

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The Product of Elimination Matrices

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