28_1 - Properties of Matrices Chapter 28 Matrix Operations...

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1 Chapter 28 Matrix Operations Properties of Matrices ± What is a matrix? ± Row vector, column vector, unit vector, zero matrix, square matrix, diagonal matrix, identity matrix, tri-diagonal matrix, upper diagonal matrix, lower diagonal matrix, permutation matrix, symmetric matrix, diagonally dominant matrix, … ± Matrix operations: addition, subtraction, multiplication, inversion, inner product of two vectors ± Vectors are linearly independent/dependent ± Determinants and ranks of a matrix, full rank matrix ± Singular matrix/non-invertible matrix ± Null vector of a matrix singular matrix ± …… Solving Linear Equations ± System of linear equations ± Solve Ax = b for x ± Existence and uniqueness ± Direct methods: find the exact solution in a finite number of steps ± Iterative methods: produce a sequence of approximate solutions hopefully converging to the exact solution Gaussian Elimination ± Solve (by elementary row operations): 39 7 4 3 15 2 3 22 3 4 2 3 2 1 3 2 1 3 2 1 = + + = + + = + + x x x x x x x x x
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28_1 - Properties of Matrices Chapter 28 Matrix Operations...

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