Graphics (INFOGR), , Block IV, lecture 8 Deb Panja. Today: Matrices. Welcome

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1 Graphics (INFOGR), , Block IV, lecture 8 Deb Panja Today: Matrices Welcome 1

2 Today Matrices: why and what? Matrix operations Determinants Adjoint/adjugate and inverse of matrices Geometric interpretation of determinants 2

3 Spatial transformations part II of the course 3

4 Spatial transformations part II of the course Why matrices? you need to execute such spatial transformations (a lot!) matrices are the vehicles you need for these tasks 4

5 Spatial transformations part II of the course Why matrices? you need to execute such spatial transformations (a lot!) matrices are the vehicles you need for these tasks That means: it s nearly impossible to dissociate matrices from transformations they achieve 5

6 Bigger scheme of things: three upcoming maths lectures 6

7 Bigger scheme of things: three upcoming maths lectures 7

8 Bigger scheme of things: three upcoming maths lectures It s been a choice to make a clean separation this way! 8

9 What is a matrix? 9

10 What is a matrix? 10

11 What is a matrix? You ll store it on a computer as a two-dimensional array: arr[3][3] 11

12 Why matrices? Example: (more about that in the next lecture) 12

13 Matrices 13

14 Matrices Dimension of the above matrix: m n when m = n, the matrix is called a square matrix a ij (i [1, m], j [1, n]) are the matrix elements/coefficients shorthand notation A = {a ij } 14

15 Matrices Dimension of the above matrix: m n when m = n, the matrix is called a square matrix a ij (i [1, m], j [1, n]) are the matrix elements/coefficients shorthand notation A = {a ij } A d-dimensional vector is a d 1 matrix an m n matrix: n vertical concatenation of m-dimensional vectors 15

16 Matrices Dimension of the above matrix: m n when m = n, the matrix is called a square matrix a ij (i [1, m], j [1, n]) are the matrix elements/coefficients shorthand notation A = {a ij } A d-dimensional vector is a d 1 matrix an m n matrix: n vertical concatenation of m-dimensional vectors A scalar is a 1 1 matrix 16

17 Special matrices Diagonal matrix: square matrix with a ij = 0 for i j; e.g., (a diagonal matrix is by definition a square matrix) 17

18 Special matrices Diagonal matrix: square matrix with a ij = 0 for i j; e.g., (a diagonal matrix is by definition a square matrix) Identity matrix: diagonal matrix with a ii = 1, e.g., (denoted by I, an identity matrix is also by definition a square matrix) 18

19 Special matrices Diagonal matrix: square matrix with a ij = 0 for i j; e.g., (a diagonal matrix is by definition a square matrix) Identity matrix: diagonal matrix with a ii = 1, e.g., (denoted by I, an identity matrix is also by definition a square matrix) Null matrix (denoted by Ø or O): a ij = 0, e.g., 19

20 Matrix operations 20

21 Addition of matrices 21

22 Addition of matrices Can only add matrices of same dimensions! 22

23 Scalar multiplication of matrices 23

24 Addition and scalar multiplication of matrices 24

25 Subtraction of matrices A and B are matrices of the same dimensions; then A B = A+( 1)B; e.g., (can only subtract matrices of same dimensions!) 25

26 Matrix multiplication A and B are m n and n p dimensional matrices respectively then C = AB is an m p-dimensional matrix 26

27 Matrix multiplication A and B are m n and n p dimensional matrices respectively then C = AB is an m p-dimensional matrix n matrix elements c ij = a ik b kj k=1 on a computer: c ij = 0; for (k = 1; k <= n; k++) c ij += a ik b kj e.g., 27

28 Vectors u = Vector dot product as a matrix multiplication u 1 u 2.. u d u v = [u 1 u 2... u d ] and v = v 1 v 2.. v d v 1 v 2.. v d = v u = u 1v 1 + u 2 v u d v d 28

29 Matrix multiplication properties In general, AB BA! (i.e., they do not commute) however, AB = BA if both A and B are diagonal in particular, IA = AI = A and AØ = ØA = Ø A(B + C) = AB + AC, (A + B)C = AC + BC: distributive (AB)C = A(BC): associative 29

30 Transpose of a matrix 30

31 Transpose of a matrix A T = transpose(a); (A T ) T = A 31

32 Determinants and cofactors (only for square matrices!) 32

33 Determinants Determinant of matrix A det(a), or A ; a scalar quantity Also as Determinants and cofactors are inextricably linked 33

34 Determinant (1 1 matrices) Determinant of a 1 1 matrix is the value of its element e.g., A = [ 5], det(a) = 5 34

35 Consider the 2 2 matrix A = Determinant (2 2 matrices) det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) [ ] a11 a 12 a 21 a 22 35

36 Consider the 2 2 matrix A = Determinant (2 2 matrices) [ ] a11 a 12 a 21 a 22 det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) = a 21 cof(a 21 ) + a 22 cof(a 22 ) 36

37 Consider the 2 2 matrix A = Determinant (2 2 matrices) [ ] a11 a 12 a 21 a 22 det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) = a 21 cof(a 21 ) + a 22 cof(a 22 ) = a 11 cof(a 11 ) + a 21 cof(a 21 ) 37

38 Consider the 2 2 matrix A = Determinant (2 2 matrices) [ ] a11 a 12 a 21 a 22 det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) = a 21 cof(a 21 ) + a 22 cof(a 22 ) = a 11 cof(a 11 ) + a 21 cof(a 21 ) = a 12 cof(a 12 ) + a 22 cof(a 22 ) 38

39 Consider the 2 2 matrix A = Determinant (2 2 matrices) det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) cof(a ij ) = ( 1) i+j det(minor(a ij )) [ ] a11 a 12 a 21 a 22 minor(a ij ) is the matrix without the i-th row and j-th column of A Q. What is the determinant of the above matrix A? 39

40 Consider the 2 2 matrix A = Determinant (2 2 matrices) det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) cof(a ij ) = ( 1) i+j det(minor(a ij )) [ ] a11 a 12 a 21 a 22 minor(a ij ) is the matrix without the i-th row and j-th column of A Q. What is the determinant of of the above matrix A? A. cof(a 11 ) = det(a 22 ) = a 22 ; cof(a 12 ) = det(a 12 ) = a 21 det(a) = a 11 a 22 a 12 a 21 ; note also that det(a) = det(a T ) Example: 40

41 Determinant (3 3 matrices) Consider the 3 3 matrix A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) + a 13 cof(a 13 ) 41

42 Determinant (3 3 matrices) Consider the 3 3 matrix A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) + a 13 cof(a 13 ) = a 21 cof(a 21 ) + a 22 cof(a 22 ) + a 23 cof(a 23 ) 42

43 Determinant (3 3 matrices) Consider the 3 3 matrix A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 det(a) = a 11 cof(a 11 ) + a 12 cof(a 12 ) + a 13 cof(a 13 ) a = a 22 a a 32 a 33 a 12 a 21 a 23 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 = a 11 a 22 a 33 a 11 a 23 a 32 + a 12 a 23 a 31 a 12 a 21 a 33 + a 13 a 21 a 32 a 13 a 22 a 31 = det(a T ) Example: 43

44 Determinant (3 3 matrices), Sarrus rule 44

45 Determinant (3 3 matrices), Sarrus rule Does not work for 2 2, 4 4,... matrices 45

46 Adjoint/adjugate and inverse (only for square matrices!) 46

47 Cofactor matrix and adjoint/adjugate Cofactor matrix C = {c ij } of matrix A = {a ij } i.e., c ij = cof(a ij ) example: Adjoint/adjugate(A) adj(a) = transpose(cof(a)) 47

48 Inverse of A A 1 = adj(a) A Inverse of a matrix Has the property AA 1 = A 1 A = I (identity matrix) Inverse does not exist if A = 0; then A is a singular matrix 48

49 Geometric interpretation of determinants 49

50 Determinants of 2 2 matrices [ ] a11 a Consider the 2 2 matrix A = 12, a 21 a 22 [ ] [ ] a11 a12 and the two vectors u = and v = a 21 a 22 then det(a) is the oriented area of the parallelogram formed by ( u, v) Oriented area is positive if u to v requires a counterclockwise rotation. Otherwise oriented area is negative. 50

51 Consider the 3 3 matrix A = and the three vectors u = Determinants of 3 3 matrices a 11 a 21 a 31 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33, v = a 12 a 22 a 23, and w = a 13 a 23 a 33 then det(a) is the oriented volume of the parallelepiped formed by ( u, v, w) Oriented volume is positive if ( u, v, w) forms a right-handed co-ordinate system. Otherwise oriented volume is negative. 51

52 Consider the 3 3 matrix A = and the three vectors u = Determinants of 3 3 matrices a 11 a 21 a 31 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33, v = a 12 a 22 a 23, and w = a 13 a 23 a 33 then det(a) is the oriented volume of the parallelepiped formed by ( u, v, w) Oriented volume is positive if ( u, v, w) forms a right-handed co-ordinate system. Otherwise oriented volume is negative. Q. How to determine ( u, v, w) s handedness? 52

53 Cross product and handedness of a co-ordinate system Right-handed co-ordinate system: ˆx ŷ = ẑ, ŷ ẑ = ˆx, ẑ ˆx = ŷ i.e., (ˆx ŷ) ẑ > 0 etc. (this is what we will use) Left-handed co-ordinate system: ˆx ŷ = ẑ, ŷ ẑ = ˆx, ẑ ˆx = ŷ i.e., (ˆx ŷ) ẑ < 0 etc. 53

54 Consider the 3 3 matrix A = and the three vectors u = Determinants of 3 3 matrices a 11 a 21 a 31 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33, v = a 12 a 22 a 23, and w = a 13 a 23 a 33 then det(a) is the oriented volume of the parallelepiped formed by ( u, v, w) Oriented volume is positive if ( u, v, w) forms a right-handed co-ordinate system. Otherwise oriented volume is negative. Q. How to determine ( u, v, w) s handedness? A. Check if ( u v) w is > < 0 54

55 Consider the n n matrix A = and the vectors u 1 = Determinants of n n matrices a 11 a 21 a 31..., u 2 = a 11 a 12 a a 21 a 22 a a 31 a 32 a a 12 a 22 a 23..., u 3 =, a 13 a 23 a 33...,... then det(a) is the oriented volume of the n-dimensional parallelepiped Q. What is the determinant when, e.g., u 3 = λ u 1 + µ u 2? 55

56 Consider the n n matrix A = and the vectors u 1 = Determinants of n n matrices a 11 a 21 a 31..., u 2 = a 11 a 12 a a 21 a 22 a a 31 a 32 a a 12 a 22 a 23..., u 3 =, a 13 a 23 a 33...,... then det(a) is the oriented volume of the n-dimensional parallelepiped Q. What is the determinant when, e.g., u 3 = λ u 1 + µ u 2? A. 0 (use the oriented volume argument!) 56

57 Summary: matrices Why? The operations defined for matrices make them special matrix dimensions, special matrices (diagonal, identity, null) Matrix operations (addition, scalar multiplication, subtraction, matrix multiplication, transpose) Determinants (only for square matrices!) Adjoint/adjugate and inverse of matrices (only for square matrices!) Geometric interpretation of determinants Next class: transformations (with matrices) 57

58 Finally, references... Book chapter 5: Linear algebra (leave out Sec. 5.4) 58