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.

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 !

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

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

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

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.

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

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

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.

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. [A B] AB = BA = In

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

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.

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

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.

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.

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

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.

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

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

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

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

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

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!

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