Introduction to Matrices in Computer Graphics

1. (Lecture # 3-4) Mathematical Concepts in Computer Graphics "Computer Graphics" Course by Qamar Abbas 1

2. What is a Matrix  A matrix is a collection of numbers arranged into a fixed number of rows and columns.  Typically the numbers are real numbers  Matrices can contain complex numbers 1 4 5 2   5 2     1 4       C C i "Computer Graphics" Course by Qamar Abbas 2

3. Row Matrix  A matrix with one row is called a row matrix. [1 2] A B  [1 3 4 5] C  [1] "Computer Graphics" Course by Qamar Abbas 3

4. Column Matrix  A matrix with one column is called a column matrix.     1 2 3     1 2 B  [1]   A       C "Computer Graphics" Course by Qamar Abbas 4

5. Square Matrix  A matrix that has equal number of rows and columns is called a square matrix           7 4 5 9 3 9 7 B  [1] C= 4 3 "Computer Graphics" Course by Qamar Abbas 5

6. Diagonal Matrix  A diagonal matrix is a square matrix with all non- diagonal elements 0.           7 0 5 0 0 0 7     1 0 0 2   A C= 0 0 "Computer Graphics" Course by Qamar Abbas 6

7. Zero Matrix  When all the elements of a matrix A are 0, we call A a 0-matrix.  We write shortly o(zero) for a 0-matrix           0 0 0 0 0 0 0      0 0 0 0 C= 0  A 0 "Computer Graphics" Course by Qamar Abbas 7

8. An identity matrix I  An identity matrix I is a diagonal matrix with all the diagonal elements = 1           1 0 1 0 0 0 1 3I C= 0  that can also be called 0     1 0 0 1   A 2I  that can also be called "Computer Graphics" Course by Qamar Abbas 8

9. Dimensions of a Matrix  The dimension of a matrix is the number of rows ans columns and it is written as rows x columns           1 2 5 8 3 6 9           1 2 5 8 3 6 9 or B= 4 B= 4 7 7 3 3  The order of this matrix is 3x3 "Computer Graphics" Course by Qamar Abbas 9

10. Matrix Addition  If two matrix are A and B then their addition is written as         1 2 3 4  4 2   A B and then 7 3               1 4 2 2 3 7 4 3   5 10 7  4 A B    "Computer Graphics" Course by Qamar Abbas 10

11. Rules for Matrix Addition  A + B = B + A  A + O = O + A = A  O + O = O  Is it true (A + B) + C = A + (B + C) ? "Computer Graphics" Course by Qamar Abbas 11

12. Matrix Negation  The negation of any matrix A is –A. If     1 2 3 4   A The          1 3 2 4   A "Computer Graphics" Course by Qamar Abbas 12

13. Matrix Subtraction  If two matrix are A and B then their addition is written as         1 2 3 4  4 2   A B and then 7 3                  1 4 2 2 3 7 4 3   3 4 0 1 A B    "Computer Graphics" Course by Qamar Abbas 13

14. Scalar multiplication of a matrix  If B is any Matrix      4 2 7 3  B  Then the scalar multiplication of B is(multiply with each number) 8 4 2 14 6       B "Computer Graphics" Course by Qamar Abbas 14

15. The transpose of a matrix  The transpose of any matrix C we change its rows to columns or change columns to rows     1 4 5 2   C  Its transpose is     1 5 4 2   t C "Computer Graphics" Course by Qamar Abbas 15

16. Matrix Multiplication  To be done in the class "Computer Graphics" Course by Qamar Abbas 16

17. Matrix Equality  Two matrix are said to be equal if they same similar elements along with the similar data values "Computer Graphics" Course by Qamar Abbas 17

18. Adjoint of a Matrix  To be done in the class "Computer Graphics" Course by Qamar Abbas 18

19. Inverse of a Matrix  To be done in the class "Computer Graphics" Course by Qamar Abbas 19

20. Symmetric Matrix  A Matrix c is called a symmetric matrix if  tc c  Show that the matrix           7 4 5 9 3 9 7 C= 4 3 "Computer Graphics" Course by Qamar Abbas 20

21. Skew-Symmetric Matrix  A Matrix c is called a skew symmetric matrix if c c   t  Show that the matrix is a skew symmetric 0 C= 2 3            2 0 0 3  0 0 "Computer Graphics" Course by Qamar Abbas 21

22. Associative and Distributive Properties  w.r.t Multiplication  A.(B.C)=(A.B).C  w.r.t Addition  A+(B+C)=(A+B)+C  Distributive Property  A(B+C) = A.B+A.C  (A+B).C = A.C+B.C "Computer Graphics" Course by Qamar Abbas 22

23. Associative and Distributive Properties  If Matrix A, B and C are as follows     1 2 3 4          4 2 1 4 5 2   A   B C  , , 7 3  Then prove the associative and distributive laws "Computer Graphics" Course by Qamar Abbas 23

24. Questions  If any Matrix B is as given           1 2 5 8 3 6 9 B= 4 7  Prove that T T (B ) = B "Computer Graphics" Course by Qamar Abbas 24

25. Questions  If any Matrix C is as follows       1 4 2 5 C= -1 C then find "Computer Graphics" Course by Qamar Abbas 25

26. How to solve System of Linear Equations?  If we have two equations 2 4    10  3 5 x x y 7 y Then solve these equations using Matrices "Computer Graphics" Course by Qamar Abbas 26