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**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.

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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 !

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**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 BC

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**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

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**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

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**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.

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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

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**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

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**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.

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**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

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

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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.

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**Solutions of linear algebraic equations**

AX = B X =A-1 B

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**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.

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**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.

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**The nontrivial solution corresponding to = 1 is:**

The eigenvalues are 1, 1, -1

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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.

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

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**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

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Homogeneous Matrix If A is an n x n constant matrix, then et 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

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**The eigenvalues are 3, 4,-2 and 6**

The associated eigenvectors are:

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**How to Solve X' = AX ? Method (1)**

Find eigenvalues of A and the corresponding eigenvectors X(1) = et Method (2) Diagonalizing A by a matrix P: Z=P-1X Z’= (P-1AP)Z X = PZ P: constant matrix

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**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!

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**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

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