Chapter Contents. A 1.6 Further Results on Systems of Equations and Invertibility 1.7 Diagonal, Triangular, and Symmetric Matrices

Transcription

1 Chapter Contents. Introduction to System of Linear Equations. Gaussian Elimination.3 Matrices and Matri Operations.4 Inverses; Rules of Matri Arithmetic.5 Elementary Matrices and a Method for Finding A.6 Further Results on Systems of Equations and Invertibility.7 Diagonal, Triangular, and Symmetric Matrices

2 . Introduction to Systems of Equations

3 Linear Equations Any straight line in y-plane can be represented algebraically by an equation of the form: a General form: define a linear equation in the n variables : Where n and b are real constants. The variables in a linear equation are sometimes called unknowns. a y,...,, a, a,..., a n, b a a... a n n b

4 Eample Linear Equations The equations 3y 7, y 3z, and 3 7 are linear. 3 4 Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or eponential functions. The equations are not linear. 3 y 5, 3 y z z 4, and y sin A solution of a linear equation is a sequence of n numbers s s,..., s n, such that the equation is satisfied. The set of all solutions of the equation is called its solution set or general solution of the equation

5 Eample Finding a Solution Set (/) Find the solution of ( a) 4 y Solution(a) we can assign an arbitrary value to and solve for y, or choose an arbitrary value for y and solve for.if we follow the first approach and assign an arbitrary value,we obtain t, y t or t, y t 4 arbitrary numbers are called parameter. for eample t t t, 3 yields thesolution 3, y as t

6 Eample Finding a Solution Set (/) Find the solution of (b) Solution(b) we can assign arbitrary values to any two variables and solve for the third variable. for eample 5 4s 7t, s, 3 t where s, t are arbitrary values

7 Linear Systems (/) A finite set of linear equations in the variables,,..., n is called a system of linear equations or a linear system. A sequence of numbers s, s,..., s n is called a solution of the system. A system has no solution is said to be inconsistent ; if there is at least one solution of the system, it is called consistent. a a a m a a a m a a... a n n mn n n n b b b An arbitrary system of m linear equations in n unknowns m

8 Linear Systems (/) Every system of linear equations has either no solutions, eactly one solution, or infinitely many solutions. A general system of two linear equations: (Figure..) a b y c ( a, b not both zero) a b ( a, b not both zero) Two lines may be parallel -> no solution Two lines may intersect at only one point -> one solution Two lines may coincide -> infinitely many solution y c

9 Augmented Matrices The location of the + s, the s, and the = s can be abbreviated by writing only the rectangular array of numbers. This is called the augmented matri for the system. Note: must be written in the same order in each equation as the unknowns and the constants must be on the right. a a a m a a a a a a m... th column m a a a m a a... a a n a a n mn b n n b mn b m n n n b b b m th row

10 Elementary Row Operations The basic method for solving a system of linear equations is to replace the given system by a new system that has the same solution set but which is easier to solve. Since the rows of an augmented matri correspond to the equations in the associated system. new systems is generally obtained in a series of steps by applying the following three types of operations to eliminate unknowns systematically. These are called elementary row operations.. Multiply an equation through by an nonzero constant. Ri->kRi. Interchange two equation. Ri <-> Rj 3. Add a multiple of one equation to another. Ri -> Ri + krj

11 Eample 3 Using Elementary row Operations = = 5 3 = R < > R

12 R -> R 3R R3 -> R3 R

13 R -> R R R3 -> R3 3R

14 R3 -> - /3R The solution 3 =, =-, = is now evident.

15 . Gaussian Elimination

16 Echelon Forms This matri which have following properties is in reduced rowechelon form (Eample, ).. If a row does not consist entirely of zeros, then the first nonzero number in the row is a. We call this a leader.. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matri. 3. In any two successive rows that do not consist entirely of zeros, the leader in the lower row occurs farther to the right than the leader in the higher row. 4. Each column that contains a leader has zeros everywhere else. A matri that has the first three properties is said to be in rowechelon form (Eample, ). A matri in reduced row-echelon form is of necessity in rowechelon form, but not conversely.

17 Eample Row-Echelon & Reduced Row-Echelon form reduced row-echelon form:, 3,, 7 4 row-echelon form: 6,,

18 Eample More on Row-Echelon and Reduced Row-Echelon form All matrices of the following types are in row-echelon form ( any real numbers substituted for the * s. ) : * * * * * * * * * * * * * *, * * * *, * * *, * * * * * * * * * * * * * * * * * * * * * * * *, * * * * *, * * * * * *, * * * * * * All matrices of the following types are in reduced rowechelon form ( any real numbers substituted for the * s. ) :

19 Eample 3 Solutions of Four Linear Systems (a) Suppose that the augmented matri for a system of linear equations have been reduced by row operations to the given reduced row-echelon form. Solve the system. (a) 5 4 Solution (a) the corresponding system of equations is : y z 5-4

20 Eample 3 Solutions of Four Linear Systems (b) (b) Solution (b). The corresponding system of equations is : leading variables free variables

21 Eample 3 Solutions of Four Linear Systems (b) We see that the free variable can be assigned an arbitrary value, say t, which then determines values of the leading variables. 3. There are infinitely many solutions, and the general solution is given by the formulas 3 4 4t, 6 t, 3t, t

22 Eample 3 Solutions of Four Linear Systems (c) (c) Solution (c). The 4th row of zeros leads to the equation places no restrictions on the solutions (why?). Thus, we can omit this equation.

23 Eample 3 Solutions of Four Linear Systems (c) Solution (c). Solving for the leading variables in terms of the free variables: The free variable can be assigned an arbitrary value,there are infinitely many solutions, and the general solution is given by the formulas s s -3t -5t, t - 4t,

24 Eample 3 Solutions of Four Linear Systems (d) (d) Solution (d): the last equation in the corresponding system of equation is 3 Since this equation cannot be satisfied, there is no solution to the system.

25 Elimination Methods (/6) We shall give a step-by-step elimination procedure that can be used to reduce any matri to reduced row-echelon form

26 Elimination Methods (/6) Step. R -> R R Step. R3 -> R3 R

27 Elimination Methods (3/6) Step3. R3 -> R3 R Step4. R3 -> /R3-4 -

28 Elimination Methods (4/6) Step5. R -> R R Step6. R -> R - R

29 Elimination Methods (5/6) Step7. R -> R + 3R3 5 - The last matri is in reduced row-echelon form.

30 Elimination Methods (6/6) Step~Step4: the above procedure produces a row-echelon form and is called Gaussian elimination. Step~Step7: the above procedure produces a reduced row-echelon form and is called Gaussian- Jordan elimination. Every matri has a unique reduced rowechelon form but a row-echelon form of a given matri is not unique.

31 Back-Substitution It is sometimes preferable to solve a system of linear equations by using Gaussian elimination to bring the augmented matri into row-echelon form without continuing all the way to the reduced row-echelon form. When this is done, the corresponding system of equations can be solved by solved by a technique called backsubstitution.

32 Eample Gaussian elimination(/) Solve by Gaussian elimination and back-substitution. Solution We convert the augmented matri to the row-echelon form The system corresponding to this matri is z y z y z y , 9, 7 7 z z y z y

33 Eample Gaussian elimination(/) Solution Solving for the leading variables 9 Substituting the bottom equation into those above Substituting the nd equation into the top y z y z y z,, 3 7 z, y,, y, z 3

34 Homogeneous Linear Systems(/) A system of linear equations is said to be homogeneous if the constant terms are all zero; that is, the system has the form : a a a m a a a m... a... a... a n n mn n n n Every homogeneous system of linear equation is consistent, since all such system have,,..., as a solution. This solution is called the trivial solution; if there are another solutions, they are called nontrivial solutions. There are only two possibilities for its solutions: The system has only the trivial solution. The system has infinitely many solutions in addition to the trivial solution. n

35 Homogeneous Linear Systems(/) In a special case of a homogeneous linear system of two linear equations in two unknowns: (fig..) a b y ( a, b a b y ( a, b not both zero) not both zero)

36 Eample 7 Gauss-Jordan Elimination(/3) Solve the following homogeneous system of linear equations by using Gauss- Jordan elimination. Solution The augmented matri Reducing this matri to reduced row-echelon form

37 Eample 7 Gauss-Jordan Elimination(/3) Solution (cont) The corresponding system of equation Solving for the leading variables is Thus the general solution is Note: the trivial solution is obtained when s=t=. t t s t s 5 4 3,,,,

38 Theorem.. A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions. Note: theorem.. applies only to homogeneous system

39 Computer Solution of Linear System Most computer algorithms for solving large linear systems are based on Gaussian elimination or Gauss-Jordan elimination. Issues Reducing roundoff errors Minimizing the use of computer memory space Solving the system with maimum speed

40 .3 Matrices and Matri Operations

41 Definition A matri is a rectangular array of numbers. The numbers in the array are called the entries in the matri.

42 Eample Eamples of matrices Some eamples of matrices Size 3, 4, 3 3,, [ ] [] 4, 3,, 3 -, 4 3 π l # rows # columns row matri or row vector column matri or column vector entries

43 Matrices Notation and Terminology(/) A general m n matri A as A = a a a M m a a a M m a a M a n n mn The entry that occurs in row i and column j of matri A will be denoted a ij or ( A) ij. If aij is real number, it is common to be referred as scalars.

44 Matrices Notation and Terminology(/) The preceding matri can be written as [ a ] or [ a ] ij m n ij A matri A with n rows and n columns is called a square matri of order n, and the shaded entries a, a, L, a nn are said to be on the main diagonal of A. a a a M m a a a M m a a M a n n mn

45 Definition Two matrices are defined to be equal if they have the same size and their corresponding entries are equal. If A = [ a ] and B = [ b ] ij then A = B if and only if ij have a ij the same size, = b ij for all i and j.

46 Eample Equality of Matrices Consider the matrices If =5, then A=B. For all other values of, the matrices A and B are not equal. There is no value of for which A=C since A and C have different sizes. = = = 4 3, 5 3, 3 C B A

47 Operations on Matrices If A and B are matrices of the same size, then the sum A+B is the matri obtained by adding the entries of B to the corresponding entries of A. Vice versa, the difference A-B is the matri obtained by subtracting the entries of B from the corresponding entries of A. Note: Matrices of different sizes cannot be added or subtracted. A + B = ( A ) + ( B ) = a + b ( ) ij ij ij ij ij ( A B ) = ( A ) ij ( B ) ij = a ij b ij ij

48 Eample 3 Addition and Subtraction Consider the matrices Then The epressions A+C, B+C, A-C, and B-C are undefined. = = =, , C B A = = , B A B A

49 Definition If A is any matri and c is any scalar, then the product ca is the matri obtained by multiplying each entry of the matri A by c. The matri ca is said to be the scalar multiple of A. In matri notation, if [ a ] A =, then ij ( ca) ij = c( A) ij = caij

50 Eample 4 Scalar Multiples (/) For the matrices We have It common practice to denote (-)B by B. = = = , 5 3 7, C B A ( ) = = = 4 3, , C B A

51 Eample 4 Scalar Multiples (/)

52 Definition If A is an m r matri and B is an r n matri, then the product AB is the m n matri whose entries are determined as follows. To find the entry in row i and column j of AB, single out row i from the matri A and column j from the matri B.Multiply the corresponding entries from the row and column together and then add up the resulting products.

53 Eample 5 Multiplying Matrices (/) Consider the matrices Solution Since A is a 3 matri and B is a 3 4 matri, the product AB is a 4 matri. And:

54 Eample 5 Multiplying Matrices (/)

55 Eamples 6 Determining Whether a Product Is Defined Suppose that A,B,and C are matrices with the following sizes: A B C Solution: Then by (3), AB is defined and is a 3 7 matri; BC is defined and is a 4 3 matri; and CA is defined and is a 7 4 matri. The products AC,CB,and BA are all undefined.

56 Partitioned Matrices A matri can be subdivided or partitioned into smaller matrices by inserting horizontal and vertical rules between selected rows and columns. For eample, below are three possible partitions of a general 3 4 matri A. The first is a partition of A into four submatrices A,A, A,and A. The second is a partition of A into its row matrices r,r, and r 3. The third is a partition of A into its column matrices c, c,c 3,and c 4.

57 Matri Multiplication by columns and by Rows Sometimes it may b desirable to find a particular row or column of a matri product AB without computing the entire product. If a,a,...,a m denote the row matrices of A and b,b,...,b n denote the column matrices of B,then it follows from Formulas (6)and (7)that

58 Eample 7 Eample5 Revisited This is the special case of a more general procedure for multiplying partitioned matrices. If A and B are the matrices in Eample 5,then from (6)the second column matri of AB can be obtained by the computation From (7) the first row matri of AB can be obtained by the computation

59 Matri Products as Linear Combinations (/)

60 Matri Products as Linear Combinations (/) In words, ()tells us that the product A of a matri A with a column matri is a linear combination of the column matrices of A with the coefficients coming from the matri. In the eercises w ask the reader to show that the product y A of a m matri y with an m n matri A is a linear combination of the row matrices of A with scalar coefficients coming from y.

61 Eample 8 Linear Combination

62 Eample 9 Columns of a Product AB as Linear Combinations

63 Matri form of a Linear System(/) m n mn m m n n n n b a a a b a a a b a a a = = = M M M M = m n mn m m n n n n b b b a a a a a a a a a M M M M = m m mn m m n n b b b a a a a a a a a a M M M M M Consider any system of m linear equations in n unknowns. Since two matrices are equal if and only if their corresponding entries are equal. The m matri on the left side of this equation can be written as a product to give:

64 Matri form of a Linear System(/) If w designate these matrices by A,,and b,respectively, the original system of m equations in n unknowns has been replaced by the single matri equation The matri A in this equation is called the coefficient matri of the system. The augmented matri for the system is obtained by adjoining b to A as the last column; thus the augmented matri is

65 Definition If A is any m n matri, then the transpose T of A,denoted by A,is defined to be the n m matri that results from interchanging the rows and columns of A ; that is, the T first column of A is the first row of A,the second column of A T is the second row of A,and so forth.

66 Eample Some Transposes (/)

67 Eample Some Transposes (/) Observe that In the special case where A is a square matri, the transpose of A can be obtained by interchanging entries that are symmetrically positioned about the main diagonal.

68 Definition If A is a square matri, then the trace of A,denoted by tr(a), is defined to be the sum of the entries on the main diagonal of A.The trace of A is undefined if A is not a square matri.

69 Eample Trace of Matri

70 .4 Inverses; Rules of Matri Arithmetic

71 Properties of Matri Operations For real numbers a and b,we always have ab=ba, which is called the commutative law for multiplication. For matrices, however, AB and BA need not be equal. Equality can fail to hold for three reasons: The product AB is defined but BA is undefined. AB and BA are both defined but have different sizes. it is possible to have AB=BA even if both AB and BA are defined and have the same size.

72 Eample AB and BA Need Not Be Equal

73 Theorem.4. Properties of Matri Arithmetic Assuming that the sizes of the matrices are such that the indicated operations can be performed, the following rules of matri arithmetic are valid:

74 Eample Associativity of Matri Multiplication

75 Zero Matrices A matri, all of whose entries are zero, such as is called a zero matri. A zero matri will be denoted by ;if it is important to emphasize the size, we shall write m n for the m n zero matri. Moreover, in keeping with our convention of using boldface symbols for matrices with one column, we will denote a zero matri with one column by.

76 Eample3 The Cancellation Law Does Not Hold Although A,it is incorrect to cancel the A from both sides of the equation AB=AC and write B=C. Also, AD=,yet A and D. Thus, the cancellation law is not valid for matri multiplication, and it is possible for a product of matrices to be zero without either factor being zero. Recall the arithmetic of real numbers :

77 Theorem.4. Properties of Zero Matrices Assuming that the sizes of the matrices are such that the indicated operations can be performed,the following rules of matri arithmetic are valid.

78 Identity Matrices Of special interest are square matrices with s on the main diagonal and s off the main diagonal, such as A matri of this form is called an identity matri and is denoted by I.If it is important to emphasize the size, we shall write I n for the n n identity matri. If A is an m n matri, then A I = A and I n m A = A Recall : the number plays in the numerical relationships a = a = a.

79 Eample4 Multiplication by an Identity Matri I m Recall : A = A and A = A, as A is an m n matri I n

80 Theorem.4.3 If R is the reduced row-echelon form of an n n matri A, then either R has a row of zeros or R is the identity matri I n.

81 Definition If A is a square matri, and if a matri B of the same size can be found such that AB=BA=I, then A is said to be invertible and B is called an inverse of A. If no such matri B can be found, then A is said to be singular. Notation: B = A

82 Eample5 Verifying the Inverse requirements

83 Eample6 A Matri with no Inverse

84 Properties of Inverses It is reasonable to ask whether an invertible matri can have more than one inverse. The net theorem shows that the answer is no an invertible matri has eactly one inverse. Theorem.4.4 Theorem.4.5 Theorem.4.6

85 Theorem.4.4 If B and C are both inverses of the matri A, then B=C.

86 Theorem.4.5

87 Theorem.4.6 If A and B are invertible matrices of the same size,then AB is invertible and AB = B A ( ) The result can be etended :

88 Eample7 Inverse of a Product

89 Definition

90 Theorem.4.7 Laws of Eponents If A is a square matri and r and s are integers,then ( r ) s rs A A r s r s A A = A +, =

91 Theorem.4.8 Laws of Eponents If A is an invertible matri,then :

92 Eample8 Powers of a Matri

93 Polynomial Epressions Involving Matrices If A is a square matri, say m m, and if n p ( ) = a + a + L+ a n () is any polynomial, then w define n p ( A) = ai + a A + L+ a n A where I is the m m identity matri. In words, p(a) is the m m matri that results when A is substituted for in () and a a is replaced by. I

94 Eample9 Matri Polynomial

95 Theorem.4.9 Properties of the Transpose If the sizes of the matrices are such that the stated operations can be performed,then Part (d) of this theorem can be etended :

96 Theorem.4. Invertibility of a Transpose If A is an invertible matri,then is also invertible and ( T ) ( A = A ) T T A

97 Eample Verifying Theorem.4.

98 .5 Elementary Matrices and a Method for Finding A -

99 Definition An n n matri is called an elementary matri if it can be obtained from the n n identity matri I n by performing a single elementary row operation.

100 Eample Elementary Matrices and Row Operations

101 Theorem.5. Row Operations by Matri Multiplication If the elementary matri E results from performing a certain row operation on and if A is an m n matri,then the product EA is the matri that results when this same row operation is performed on A. I m When a matri A is multiplied on the left by an elementary matri E,the effect is to performan elementary row operation on A.

102 Eample Using Elementary Matrices

103 Inverse Operations If an elementary row operation is applied to an identity matri I to produce an elementary matri E,then there is a second row operation that, when applied to E, produces I back again. Table.The operations on the right side of this table are called the inverse operations of the corresponding operations on the left.

104 Eample3 Row Operations and Inverse Row Operation The identity matri to obtain an elementary matri E,then E is restored to the identity matri by applying the inverse row operation.

105 Theorem.5. Every elementary matri is invertible,and the inverse is also an elementary matri.

106 Theorem.5.3 Equivalent Statements If A is an n n matri,then the following statements are equivalent,that is,all true or all false.

107 Row Equivalence Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent. With this terminology it follows from parts (a )and (c ) of Theorem.5.3 that an n n matri A is invertible if and only if it is row equivalent to the n n identity matri.

108 A method for Inverting Matrices

109 Eample4 Using Row Operations to Find (/3) Find the inverse of 3 A = Solution: To accomplish this we shall adjoin the identity matri to the right side of A,thereby producing a matri of the form [ A I ] we shall apply row operations to this matri until the left side is reduced to I ;these operations will convert the right side to A,so that the final matri will have the form [ ] I A A

110 Eample4 Using Row Operations to Find (/3) A

111 Eample4 Using Row Operations to Find (3/3) A

112 Eample5 Showing That a Matri Is Not Invertible

113 Eample6 A Consequence of Invertibility

114 .6 Further Results on Systems of Equations and Invertibility

115 Theorem.6. Every system of linear equations has either no solutions,eactly one solution,or in finitely many solutions. Recall Section. (based on Figure.. )

116 Theorem.6. If A is an invertible n n matri,then for each n matri b,the system of equations A =b has eactly one solution,namely, = A b.

117 Eample Solution of a Linear System Using A

118 Linear systems with a Common Coefficient Matri one is concerned with solving a sequence of systems A = b A = b, A = b,, A =, 3 L Each of which has the same square coefficient matri A.If A is invertible, then the solutions = A b, = A b, = A b, L = A A more efficient method is to form the matri By reducing()to reduced row-echelon form we can solve all k systems at once by Gauss-Jordan elimination. This method has the added advantage that it applies even when A is not invertible. b k 3 3, [ A b b L ] b k k b k

119 Eample Solving Two Linear Systems at Once Solve the systems Solution

120 Theorem.6.3 Up to now, to show that an n n matri A is invertible, it has been necessary to find an n n matri B such that AB=I and BA=I We produce an n n matri B satisfying either condition, then the other condition holds automatically.

121 Theorem.6.4 Equivalent Statements

122 Theorem.6.5 Let A and B be square matrices of the same size. If AB is invertible,then A and B must also be invertible.

123 Eample3 Determining Consistency by Elimination (/)

124 Eample3 Determining Consistency by Elimination (/)

125 Eample4 Determining Consistency by Elimination(/)

126 Eample4 Determining Consistency by Elimination(/)

127 .7 Diagonal, Triangular, and Symmetric Matrices

128 Diagonal Matrices (/3) A square matri in which all the entries off the main diagonal are zero is called a diagonal matri. Here are some eamples. A general n n diagonal matri D can be written as

129 Diagonal Matrices (/3) A diagonal matri is invertible if and only if all of its diagonal entries are nonzero; Powers of diagonal matrices are easy to compute; we leave it for the reader to verify that if D is the diagonal matri () and k is a positive integer, then:

130 Diagonal Matrices (3/3) Matri products that involve diagonal factors are especially easy to compute. For eample, To multiply a matri A on the left by a diagonal matri D, one can multiply successive rows of A by the successive diagonal entries of D, and to multiply A on the right by D one can multiply successive columns of A by the successive diagonal entries of D.

131 Eample Inverses and Powers of Diagonal Matrices

132 Triangular Matrices A square matri in which all the entries above the main diagonal are zero is called lower triangular. A square matri in which all the entries below the main diagonal are zero is called upper triangular. A matri that is either upper triangular or low r triangular is called triangular.

133 Eample Upper and Lower Triangular Matrices

134 Theorem.7.

135 Eample3 Upper Triangular Matrices

136 Symmetric Matrices A square matri A is called symmetric if T A=. A The entries on the main diagonal may b arbitrary, but mirror images of entries across the main diagonal must be equal. [ ] a matri A = a ij is symmetric if and only if a = for all values of i and j. ij a ji

137 Eample4 Symmetric Matrices

138 Theorem.7. T ( ) BA T T Recall : AB = B A = Since AB and BA are not usually equal, it follows that AB will not usually be symmetric. However, in the special case where AB=BA,the product AB will be symmetric. If A and B are matrices such that AB=BA,then we say that A and B commute. In summary: The product of two symmetric matrices is symmetric if and only if the matrices commute.

139 Eample5 Products of Symmetric Matrices The first of the following equations shows a product of symmetric matrices that is not symmetric, and the second shows a product of symmetric matrices that is symmetric. We conclude that the factors in the first equation do not commute, but those in the second equation do.

140 Theorem.7.3 If A is an invertible symmetric matri,then is symmetric. A In general, a symmetric matri need not be invertible.

141 Products A T A and AA T T T The products A A and AA are both square matrices. T ---- the matri AA has size m m and the matri A T has size n n. Such products are always symmetric since A

142 Eample6 The Product of a Matri and Its Transpose Is Symmetric

143 Theorem.7.4 A is square matri. If A is an invertible matri,then AA and A T A are also invertible. T