化工應用數學

1. 化工應用數學 授課教師： 郭修伯 Lecture 9 Matrices Consideration a greater numbers of variables as a single quantity called a matrix.

2. Matrices • We can store objects (numbers, functions …) in named locations/grids. • A matrix has n rows and m columns. A is “n by m”. Each element is called aij. • The element of matrix product AB aij= i, j element = < row i of A > • < column j of B > Think of the vector product !

3. Differences between Matrix Operations and Real Number Operations • Matrix multiplication in not commutative. • There is in general no “cancellation” of A in an equation AB = AC • The product AB may be a zero matrix with neither A nor B a zero matrix. AB BA AB = AC, but BC

4. Matrices • What do we need to know about matrices? • square matrix • the number of rows of elements is equal 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 all the diagonal elements are all unity • the transpose matrix • A (n x m)  A’ (m x n) • If AA’ = I, the matrix A is “orthogonal” • the transpose of the product of two matrices is equal to the product of their transposes taken in the reverse order: (AB)’ = B’A’ • symmetric matrix

5. Matrices • Elementary row operations • interchange of two rows • Multiplication of a row by a nonzero scalar • Addition of a scalar multiple of one row to another row • Any elementary row operation on an n x m matrix A can be achieved by multiplying A on the left by the elementary matrix formed by performing the same row operation on In (unit matrix). EA = B

6. The Reduced Form of a Matrix • A is a reduced matrix if • the leading entry of any nonzero row is 1 • a row has its leading entry in column c, all other elements of column are zero • each row having all zero elements lies below any row having a nonzero element • the leading entry in row r1 lies in column c1 and the leading entry of row r2 is in column c2, and r1 < r2, then c1 < c2.

7. The rank of a Matrix • rank (A) = number of nonzero rows of the reduced form of a matrix A = dimension of the row space of A. • The row space of A means all the linear combinations of the row vectors of A. • The row vectors of A are: F1 = < -1,4,0,1,6 > and F2 = < -2,8,0,2,12 > • The row space of A is the subspace of R5 consisting of all linear combinations: F1+F2 • rank (A) = 1

8. The Determinant of a Square Matrix • A number produced from the matrix A: • It is defined as a sum of multiples of (n-1) x (n-1) determinants formed from the elements of A. • The cofactor (or Laplace) expansion of |A| by row k is defined to be the sum of the element of row k, each multiplied by its cofactor: | A |, or det (A) Mkj is the minor of akj in A

9. The Determinant of a Square Matrix • If B is formed from A by multiplying any row or column of A by a scalar , |B| = |A|. • If A has a zero row or column, |A| = 0. • If B is obtained from A by interchanging two rows or columns, |B| = -|A|. • If two rows or columns of A are identical, |A| = 0. • If one row (or column) is a constant multiple of another, |A| = 0. • Suppose we obtained B from A by adding a constant multiple of one row (or column) to another row (or column). Then |B| = |A|. • For any square matrix A, |A| = |At|. • If A and B are n x n matrices, |AB| = |A||B|. • If U = [uij] is upper triangular, |U| = u11u22…unn.

10. [A B] Matrix • If AX = B, then the augmented matrix is: • If A and B are n x n matrices, we call each other an inverse of the other if • A square matrix is called nonsingular when it has an inverse and singular when it does not. AB = BA = In

11. Inverse Matrix • How to find A-1 ? • Method (1) • Method (2) • Why find A-1 ?

12. Cramer’s Rule • If A is an n x n nonsingular matrix, the unique solution of the nonhomogeneous system AX = B is given by X =A-1 B • solve A(k; B) is the n x n matrix obtained by replacing column k of A with B.

13. Solutions of linear algebraic equations AX = B X =A-1 B

14. Eigenvalues and Eigenvectors • If A is an n x n matrix, a real or complex number  is called an eigenvalue of A if, for some nonzero n x 1 matrix X, • Any nonzero n x 1 matrix X satisfying this equation for some number  is called an eigenvector of A associated with the eigenvalue . • An n x n matrix has exactly n eigenvalues. • Eigenvectors associated with distinct eigenvalues of a matrix are linearly independent.

15. Eigenvalues • If A is an n x n matrix, then •  is an eigenvalue of A if and only if | In-A | = 0. • if  is an eigenvalue of A, any nontrivial solution of (In-A)X = 0 is an eigenvector of A associated with . • How to find the eigenvalues of A? • Solving the characteristic equation of A : (In-A)X = 0 • The eigenvalues of a diagonal matrix are its main diagonal elements.

16. The nontrivial solution corresponding to  = 1 is: The nontrivial solution corresponding to  = -1 is: The eigenvalues are 1, 1, -1

17. Diagonal Matrix • The eigenvalues of a diagonal matrix are its main diagonal elements. • An n x n matrix is diagonalizable if there exist an n x n matrix P such that P-1AP is a diagonal matrix. • The Matrix P is composed by the eigenvectors of A • NOT every matrix is diagonalizable. If A does not have n linearly independent eigenvectors, A is not diagonalizable. • Any n x n matrix with n distinct eigenvalues is diagonalizable.

18. The eigenvalues are 1, -1, -2 The associated eigenvectors are:

19. Matrix Solution of Systems of Differential Equations • Best advantage: Solve many differential equations simutaneously! • A fundamental matrix for the system X' = AX has columns consisted of the linearly independent solutions. • If  is the fundamental matrix for X' = AX on the interval J, then the general solution of X' = AX is X = C, where C is an n x 1 matrix of arbitrary constants. • Let  be any solution of X' = AX + G, then the general solution of X' = AX + G is  = C +  two independent solutions

20. Homogeneous Matrix • If A is an n x n constant matrix, then et is a nontrivial solution of X' = AX if and only if  is an eigenvalue of A and  is a corresponding eigenvector. • If  =  + i is an eigenvalue of A, with a corresponding eigenvector  = U + iV, then two linealy independent solutions of X' = AX are: and The eigenvalues are 1, 6 The associated eigenvectors are: X = C

21. The eigenvalues are 3, 4,-2 and 6 The associated eigenvectors are:

22. How to Solve X' = AX ? • Method (1) • Find eigenvalues of A and the corresponding eigenvectors • X(1) = et • Method (2) • Diagonalizing A by a matrix P: Z=P-1X • Z’= (P-1AP)Z • X = PZ P: constant matrix

23. How to Solve X' = AX + G ? • Diagonalizing A by a matrix P • Z’= (P-1AP)Z + P-1G • X = PZ How about matrix A which is not diagonalizable? (i.e. does not have n linearly independent eigenvectors) exponential matrix!

24. Exponential Matrix • Define • Procedure to find solutions of X' = AX: • find eigenvalues of A (which is not diagonalizable) • find C, let (A-I)k C = 0 and (A- I)k-1  0 • A solusion is then eAtC = • General solution for X' = AX + G: k=1 k=2