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Matrices
- 2. Overview of the lecture ● Matrices ● Matrix algebra ● Vectors ● Homogeneous coordinates ● Transformations in homogeneous coordinates
- 3. Matrices Matrix is an array of numbers − − 05565 08707 43601 1341
- 4. Matrices Matrix is an array of numbers −− − −− 3.81109.09.0 1.951.33.90 84.119.02.3
- 5. Matrices Matrix is an array of numbers 3231 2221 1211 aa aa aa
- 6. Matrices Matrix is an array of numbers − 5.0 2.9 8.7 2.1 5.0
- 7. Terminology A row of the matrix − − 05565 08707 43601 1341
- 8. Terminology A column of the matrix − − 05565 08707 43601 1341
- 9. Terminology An element of the matrix − − 05565 08707 43601 1341
- 10. Terminology If a matrix has m rows and n columns we say that its dimension is m x n. The dimension of this matrix is 3 x 5 −− − −− 3.81109.09.0 1.951.33.90 84.119.02.3
- 11. Addition The addition of matrices is component-wise. The two matrices must have the same dimension. ++ ++ = + 22222121 12121111 2221 1211 2221 1211 baba baba bb bb aa aa
- 13. Multiplication by scalar The multiplication of a matrix by a scalar (by a number) is again component-wise. ⋅⋅ ⋅⋅ = ⋅ 2221 1211 2221 1211 acac acac aa aa c
- 14. Multiplication by scalar Example 2.2: = ⋅ 60 213 20 71 3
- 15. Let A be a n x m matrix and let B be a m x k matrix. Then A*B is a n x k matrix Matrix multiplication = nknn k k mkmm k k nmnn m m ccc ccc ccc bbb bbb bbb aaa aaa aaa 21 22221 11211 21 22221 11211 21 22221 11211 ∑= = m l ljilij bac 1
- 16. Matrix multiplication Example 2.3: = 15 113 011 200 210 125 431
- 17. Matrix multiplication Example 2.3: = 715 113 011 200 210 125 431
- 18. Matrix multiplication Example 2.3: = 6715 113 011 200 210 125 431
- 21. Short review of matrices Elementary transformations (keep order and rank) 1-Swap (Interchange) )()( jrowirowRij ⇔= )()( jcolumnicolumnCij ⇔= 2-Multiply )(.)( irowkirowkRi ⇔= 0k ≠ )(.)( icolumnkicolumnkCi ⇔= 3-addition 0k ≠ )(.)()( jrowkirowirowkRR ji +⇔=+ )(.)()( jcolumnkicolumnicolumnkCC ji +⇔=+
- 22. Rank of Matrices. • It is the number of nonzero rows of the echelon form.
- 23. Short review of matrices Transpose Rows and columns of an mXn matrix are interchanged To form the nXm matrix = 434241 333231 232221 131211 aaa aaa aaa aaa A = 43332313 42322212 41312111 aaaa aaaa aaaa 'A 'B'A)'BA( +=+ 'A'B)'AB( =
- 24. Short review of matrices Null matrix: O A0OAAO ∀== Orthogonal matrix I'AAA'A == θθ θ−θ = )(Cos)(Sin )(Sin)(Cos A θθ− θθ = )(Cos)(Sin )(Sin)(Cos 'A = 10 01 'AA
- 25. • Homogeneous equations: AX=0 1. unique solution 2. Infinite solution • Nonhomogeneous equations: AX=B 1. No solution 2. unique solution 3. Infinite solution
- 26. Overview of the lecture ● Matrices ● Matrix algebra ● Vectors ● Homogeneous coordinates ● Transformations in homogeneous coordinates
- 27. Vectors A column vector is a n x 1 matrix A row vector is a 1 x n matrix 1 21 11 na a a [ ]naaa 11211
- 28. Vectors The point (x , y , z) of the 3D space can be written as a column vector or a row vector z y x [ ]zyx
- 29. Vectors Example 2.4: = 3 0 1 210 125 431
- 33. Applications of matrices • Solutions of simultaneous linear Homogeneous and Nonhomogeneous equations. • Linear dependence and independence • Balancing the chemical reactions. • Determine the current in the circuit. • Array used in programming.
- 34. • Matrix transformations are used in computer graphics (CAD). • Store the data. • Cryptography (Secrete writing). • New view of complex number. Applications of matrices
- 35. Overview of the lecture ● Matrices ● Matrix algebra ● Vectors ● Homogeneous coordinates ● Transformations in homogeneous coordinates
- 36. Homogenous coordinates • Why don’t we stick with the usual (Cartesian) coordinates ? • In the previous lecture, we noticed that the equations of the translation are not (can not) be described by a multiplication by a 3x3 matrix. • A unified matrix description of translation, rotation, scaling (and other transformations we do not cover here) requires the use of 4x4 matrices. • This uniformity is especially useful when we want to compute combinations of transformations. • In the beginning, it is difficult to appreciate the advantages of homogenous coordinates, but they are a standard in graphics programming.
- 37. Homogeneous coordinates A point of the 3D space is written in homogeneous coordinates as z y x ⋅ ⋅ ⋅ w zw yw xw 0≠w
- 38. Homogeneous coordinates Notice that the homogenous coordinates of a 3D point are not unique. In many applications it is sufficient (and convenient) to fix w = 1, in which case the homogenous coordinates become 1 z y x
- 39. Overview of the lecture ● Matrices ● Matrix algebra ● Vectors ● Homogeneous coordinates ● Transformations in homogeneous coordinates
- 40. Transformations in homogeneous coordinates • Translation • Rotation • Scaling
- 41. Translation A translation moves the object along a vector ),,, 1( 000 zyx )1,0,0,0( x z y
- 42. Translation In homogeneous coordinates the translation by has equation ),,( 000 zyx = 1 ' ' ' 11000 100 010 001 0 0 0 z y x z y x z y x
- 43. Rotation
- 44. Rotation In homogeneous coordinates, the rotation through θ along the x-axis is given by = − 1 ' ' ' 11000 0cossin0 0sincos0 0001 z y x z y x θθ θθ
- 45. Rotation the equation of the rotation through θ along the y-axis is = − 1 ' ' ' 11000 0cos0sin 0010 0sin0cos z y x z y x θθ θθ
- 46. Rotation the equation of the rotation through θ along the z-axis is = − 1 ' ' ' 11000 0100 00cossin 00sincos z y x z y x θθ θθ
- 47. Scaling The equation of the uniform scaling by s is Uniform scaling = 1 ' ' ' 11000 000 000 000 z y x z y x s s s
- 48. Scaling The general equation of the scaling is General scaling = 1 ' ' ' 11000 000 000 000 3 2 1 z y x z y x s s s
- 49. Discussion • Translations and rotations are the most common transformations of 3D space. • They do not alter the shape of an object, but only change its position in space. • The distance between two points is the same before and after a translation or rotation. • In a mathematical language, we say that translations and rotations are isometries. • Translations and rotations preserve the orientation of an object.
- 50. Discussion An example of a non-orientation preserving isometry is the reflection.