化工應用數學

1. 化工應用數學 Matrices 授課教師： 林佳璋

2. Matrix matrix Each symbol such as a11, a2n, or amn is called an element of the matrix. The square bracket enclosing the array signifies that it is a matrix. The two straight lines enclosing a square array signify that to be a determinant. The matrix A contains m rows and n columns. A is called an m by n matrix, or A is said to be of order “m×n”. 2 by 3 matrix implying that there are two rows and three columns

3. Type of Matrix row matrix- having n element arranged in a single row column matrix- having n element arranged in a single column unit column matrix

4. Type of Matrix square matrix- having the number of rows of elements equivalent to the number of columns of elements. diagonal matrix- all the elements except those in the diagonal from the top left-hand corner to the bottom right-hand corner are zero. unit matrix- a diagonal matrix in which the diagonal elements are all unity. unit matrix of order three null matrix- all elements in matrix are zero, and is denoted by 0.

5. Matrix Algebra Matrix Addition 1.This operation can only be carried out on matrices of the same order. 2.The sum of two matrices is obtained by adding together the corresponding elements of each matrix. Ex: Note:the sum or difference does not exist if the matrices are of different order

6. Matrix Algebra Scalar Multiplication If any matrix is multiplied by a number, the elements of the matrix so formed are the products of the number and the corresponding elements of the original matrix. Matrix Multiplication The product of two matrices will exist only when the number of columns of the first matrix is equal to the number of rows in the second matrix. When this condition is satisfied the two matrices are said to be “conformable” and they yield a product.

7. Matrix Algebra If the matrix A is of order (m×n) and the matrix B is of order (n×m) the product C of AB=C is a matrix of order (m×n).(n×m)=(m×m). On the other hand, the product D of BA=D is matrix of order (n×m).(m×n)=(n×n). (i) C is obtained by pre-multiplying B by A. (ii) C is obtained by post-multiplying A by B.

8. Matrix Algebra Matrix Multiplication In the exceptional case when the products of two square matrices are equal, i.e. AB=BA, the matrices are said to commute, or be commutable. Note:

9. Determinants of Square Matrices and Matrix Products The determinant of the product of two square matrices is equal to the product of their determinants. Even though in general ABBA Note: If A is a square matrix and , the matrix is called a “singular matrix”. On the other hand, if , the matrix is “non-singular”.

10. Transpose of a Matrix If the rows of an (n×m) matrix are written in the form of columns, a new matrix of order (m×n) will be formed. This new matrix is called the “transpose” of the original matrix. Note: A matrix and its transpose are always conformable for multiplication and if AA’=I, the matrix A is said to be “orthogonal”. Note: The transpose of the product of two matrices is equal to the product of their transposes taken in the reverse order.

11. Adjoint Matrices If aij is any element of a sqaure matrix of order (nn), and the cofactor of aij in the determinant is Aij, the transpose of the matrix whose elements are made up of all the Aijs is called the “adjoint of A”, and is written as

12. Inverse of a Square Matrix Consider a square matrix A to be non-singular and of order (nn) Define a new matrix

13. Inverse of a Matrix

14. Inverse of a Matrix

15. Solution of Linear Algebraic Equations

16. Solution of Linear Algebraic Equations x1=5.704, x2=0.261, x3=1.452, x4=1.313

17. Matrix Series Powers of Matrices Matrix Polynomials

18. Matrix Series X is a square matrix

19. Differentiation and Integration of Matrices Differentiation Integration

20. Lambda-Matrices A -matrix is a square matrix in which the elements are function of a scalar parameter . when =1 the determinant of the matrix () is zero, and hence this matrix is singular. However, its minors of order two are non- vanishing, therefore the -matrix is of rank 2.

21. Characteristic Equation Let A be a square matrix of order n whose elements are all constants, and let x be a column whose elements are related by the equation where  is a scalar coefficient. where (A-I) is a square -matrix of order n. characteristic equation of matrix A Roots p are called the “latent roots” or the “eigenvalues” of A. Note: A and its transpose have the same latent roots.

22. Diagonal Canonical Form The vector corresponding to the latent root is called a “latent column vector”. Let k1, k2, k3, …,kn be the latent column vectors corresponding to the latent roots of A, and K the square matrix comprising all the column vectors. K will be of order n and non-singular. Showing that the matrix A can be reduced to what is known as “diagonal canonical form”

23. Example Evaluate the latent roots and reduce the following matrix to diagonal form

24. Example

25. Exponents of Matrix

26. Exponents of Matrix

27. Cayley-Hamilton Theorem Any square matrix satisfies its own characteristic equation. The characteristic equation of the above matrix is

28. Cayley-Hamilton Theorem Calculate A4 if The characteristic equation of the matrix A is Replacing  by A and rearranging gives Multiplying by A gives Multiplying by A gives

29. Example Consider the matrix Find (a) A1/2=? (b) sinA=? (c) exp(A2)=?

30. Example

31. Application of Matrix System of first order linear ordinary differential equation n=1 n>1

32. Example Solve

33. Example Solve

34. Example

35. Application of Matrix Consider system of first order non-linear ordinary differential equation Let

36. Example Solve

37. Example

38. Application of Matrix System of second order homogeneous ordinary differential equation Let

39. Example Solve Let

40. Example