lecture05 - Lecture Lecture 5 2.1 Introduction to Matrices...

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ecture 5 Lecture 5 1 troduction to Matrices 2.1 Introduction to Matrices 2.2 Matrix Operations Chapter 2 Matrices 1
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Announcement ± Official Tutorial/Lab groups ² Stick to your official groups throughout the semester ² o more changes allowed for tutorial/lab group No more changes allowed for tutorial/lab group ± First tutorial session this week ± First lab session next week ² Lab worksheets in workbin ² Print out the worksheets and bring to the lab Lecture 5 Announcement 2
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ection 2 1 Section 2.1 Introduction to Matrices Objective hat are the ize ntries rder f a matrix? • What are the size , entries , order of a matrix? •Whatare diagonal , identity , symmetric , triangular matrices? • How to express matrices using ( i, j )-entries? Chapter 2 Matrices 3
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1, 2)- ntry What are the size and entries of a matrix? Matrices (summary) n a a a " 1 12 11 (1, 2) entry (2, 1)-entry row = n a a a # # # " 2 22 21 A can be simplified as A = ( a ij ) m x n or ( a ij ) “zipped” form mn m m a a a " 2 1 column “unzipped” form number of rows is m number of columns is n e say The ize f the matrix a ij denotes the number in the i th row and j th column. We say : The size of the matrix A is m x n A is an m x n matrix Chapter 2 Matrices 4 : a ij is the ( i , j )-entry of the matrix A
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What are the size and entries of a matrix? Example 2.1.6 1. A = ( a ) where a = i + j ij 2 x 3 ij = A + = even is if 1 j i b 2. B = ( b ) x where + odd is if 1 j i ij ij 3 2 = B Chapter 2 Matrices 5 Learn how to describe various types of matrices in terms of ( i, j )-entries
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What are the order and diagonal of a square matrix? Square matrices (summary) ⎛⎞ " 1 12 1 aa a on- iagonal ⎜⎟ = " ## % # 11 21 22 2 n n a A non diagonal entries of A ⎝⎠ " 12 nn n n a diagonal of A same number of rows and columns A is an n x n matrix A = ( a ij ) is a square matrix of order n a 11 , a 22 , …, a nn are called the diagonal entries Chapter 2 Matrices 6 a ij , i j , are called the non-diagonal entries
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What are diagonal, scalar, identity matrices? How to express them using ( i, j )-entries? Types of square matrices (summary) Diagonal ll on- iagonal 00 matrix all non diagonal entries are zero ⎛⎞ ⎜⎟ 100 030 02 a ij = 0 whenever i j Scalar matrix diagonal matrix with all diagonal ntries e ame ⎝⎠ 002 300 30 if 0 ij entity entries the same iagonal matrix 003 = = if ij a ci j Identity matrix diagonal matrix with all diagonal entries equal 1 010 = = i 0 1 f if ij a I n Chapter 2 Matrices 7 001
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What are symmetric and triangular matrices? How to express them using ( i, j )-entries? Types of square matrices (summary) Zero all entries equal ⎛⎞ 000 matrix q to zero can be non-square ⎜⎟ ⎝⎠ a ij = 0 for all i, j 0 m x n 1 0 Symmetric matrix k th row “equal” k th column for all k a ij = a ji for all i , j 11 132 02 2 Upper triangular matrix all entries below diagonals are zero 122 033 002 a ij = 0 for all i > j Lower triangular matrix all entries above diagonals are zero 100 230 32 a ij = 0 for all i < j
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lecture05 - Lecture Lecture 5 2.1 Introduction to Matrices...

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