Linear Algebra, Theory And Applications, 2012a

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44 MATRICES AND LINEAR TRANSFORMATIONS from location 2 to location 3 although it is possible to go from location 3 to location 2. Lets refer to moving along one of these directed lines as a step. The following 3 × 3 matrix is a numerical way of writing the above graph. This is sometimes called a digraph, short for directed graph. ⎛ ⎝ 1 1 1 0 1 0 ⎞ ⎠ 1 1 0 Thus a ij , the entry in the i th row and j th column represents the number of ways to go from location i to location j in one step. Problem: Find the number of ways to go from i to j using exactly k steps. Denote the answer to the above problem by a k ij . We don’t know what it is right now unless k = 1 when it equals a ij described above. However, if we did know what it was, we could find a k+1 ij as follows. a k+1 ij = ∑ a k ira rj r This is because if you go from i to j in k + 1 steps, you first go from i to r in k steps and then for each of these ways there are a rj ways to go from there to j. Thus a k ir a rj gives the number of ways to go from i to j in k + 1 steps such that the k th step leaves you at location r. Adding these gives the above sum. Now you recognize this as the ij th entry of the product of two matrices. Thus a 2 ij = ∑ r a ir a rj , a 3 ij = ∑ r a 2 ira rj and so forth. From the above definition of matrix multiplication, this shows that if A is the matrix associated with the directed graph as above, then a k ij is just the ijth entry of A k where A k is just what you would think it should be, A multiplied by itself k times. Thus in the above example, to find the number of ways of going from 1 to 3 in two steps you would take that matrix and multiply it by itself and then take the entry in the first row and third column. Thus ⎛ ⎝ 1 1 1 1 0 0 1 1 0 ⎞ ⎠ 2 = ⎛ ⎝ 3 2 1 1 1 1 2 1 1 and you see there is exactly one way to go from 1 to 3 in two steps. You can easily see this is true from looking at the graph also. Note there are three ways to go from 1 to 1 in 2 steps. Can you find them from the graph? What would you do if you wanted to consider 5 steps? ⎛ ⎝ 1 1 1 1 0 0 1 1 0 ⎞ ⎠ 5 ⎛ = ⎝ 28 19 13 13 9 6 19 13 9 There are 19 ways to go from 1 to 2 in five steps. Do you think you could list them all by looking at the graph? I don’t think you could do it without wasting a lot of time. Of course there is nothing sacred about having only three locations. Everything works just as well with any number of locations. In general if you have n locations, you would need to use a n × n matrix. Example 2.1.13 Consider the following directed graph. ⎞ ⎠ ⎞ ⎠
2.1. MATRICES 45 1 2 3 4 Write the matrix which is associated with this directed graph and find the number of ways to go from 2 to 4 in three steps. Here you need to use a 4×4 matrix. The one you need is ⎛ ⎞ 0 1 1 0 ⎜ 1 0 0 0 ⎟ ⎝ 1 1 0 1 ⎠ 0 1 0 1 Then to find the answer, you just need to multiply this matrix by itself three times and look at the entry in the second row and fourth column. ⎛ ⎞3 ⎛ ⎞ 0 1 1 0 1 3 2 1 ⎜ 1 0 0 0 ⎟ ⎝ 1 1 0 1 ⎠ = ⎜ 2 1 0 1 ⎟ ⎝ 3 3 1 2 ⎠ 0 1 0 1 1 2 1 1 Thereisexactlyonewaytogofrom2to4inthreesteps. How many ways would there be of going from 2 to 4 in five steps? ⎛ ⎞5 ⎛ ⎞ 0 1 1 0 5 9 5 4 ⎜ 1 0 0 0 ⎟ ⎝ 1 1 0 1 ⎠ = ⎜ 5 4 1 3 ⎟ ⎝ 9 10 4 6 ⎠ 0 1 0 1 4 6 3 3 There are three ways. Note there are 10 ways to go from 3 to 2 in five steps. This is an interesting application of the concept of the ij th entry of the product matrices. 2.1.3 Properties Of Matrix Multiplication As pointed out above, sometimes it is possible to multiply matrices in one order but not in the other order. What if it makes sense to multiply them in either order? Will they be equal then? Example 2.1.14 Compare ( 1 2 3 4 The first product is ( 1 2 3 4 the second product is ( 0 1 1 0 )( 0 1 1 0 )( 0 1 1 0 )( 1 2 3 4 ) and ) = ) = ( 0 1 1 0 ( 2 1 4 3 ( 3 4 1 2 ) , )( 1 2 3 4 and you see these are not equal. Therefore, you cannot conclude that AB = BA for matrix multiplication. However, there are some properties which do hold. ) , ) .
Page 1 and 2: Linear Algebra, Theory And Applicat
Page 3 and 4: Contents 1 Preliminaries 11 1.1 Set
Page 5 and 6: CONTENTS 5 8 Vector Spaces And Fiel
Page 7 and 8: CONTENTS 7 E The Fundamental Theore
Page 9 and 10: Preface This is a book on linear al
Page 11 and 12: Preliminaries 1.1 Sets And Set Nota
Page 13 and 14: 1.3. THE NUMBER LINE AND ALGEBRA OF
Page 15 and 16: 1.5. THE COMPLEX NUMBERS 15 Also fr
Page 17 and 18: 1.5. THE COMPLEX NUMBERS 17 and so
Page 19 and 20: 1.6. EXERCISES 19 Eventually, for r
Page 21 and 22: 1.8. WELL ORDERING AND ARCHIMEDEAN
Page 23 and 24: 1.9. DIVISION AND NUMBERS 23 Theore
Page 25 and 26: 1.9. DIVISION AND NUMBERS 25 Exampl
Page 27 and 28: 1.10. SYSTEMS OF EQUATIONS 27 Why i
Page 29 and 30: 1.10. SYSTEMS OF EQUATIONS 29 The a
Page 31 and 32: 1.11. EXERCISES 31 It follows x =
Page 33 and 34: 1.14. EXERCISES 33 the existence of
Page 35 and 36: 1.15. THE INNER PRODUCT IN F N 35 s
Page 37 and 38: Matrices And Linear Transformations
Page 39 and 40: 2.1. MATRICES 39 Definition 2.1.2 M
Page 41 and 42: 2.1. MATRICES 41 is it possible to
Page 43: 2.1. MATRICES 43 a3× 3 matrix as d
Page 47 and 48: = ∑ k ( B T ) ik ( A T ) kj 2.1.
Page 49 and 50: 2.1. MATRICES 49 As described above
Page 51 and 52: 2.2. EXERCISES 51 Write the augment
Page 53 and 54: 2.3. LINEAR TRANSFORMATIONS 53 (e)
Page 55 and 56: 2.3. LINEAR TRANSFORMATIONS 55 Proo
Page 57 and 58: 2.4. SUBSPACES AND SPANS 57 Proof:
Page 59 and 60: 2.4. SUBSPACES AND SPANS 59 Proof 3
Page 61 and 62: 2.5. AN APPLICATION TO MATRICES 61
Page 63 and 64: 2.6. MATRICES AND CALCULUS 63 2.6.1
Page 65 and 66: 2.6. MATRICES AND CALCULUS 65 where
Page 67 and 68: 2.6. MATRICES AND CALCULUS 67 There
Page 69 and 70: 2.6. MATRICES AND CALCULUS 69 Using
Page 71 and 72: 2.7. EXERCISES 71 this will suffice
Page 73 and 74: 2.7. EXERCISES 73 22. If A is a lin
Page 75 and 76: 2.7. EXERCISES 75 35. ↑Suppose yo
Page 77 and 78: Determinants 3.1 Basic Techniques A
Page 79 and 80: 3.1. BASIC TECHNIQUES AND PROPERTIE
Page 81 and 82: 3.2. EXERCISES 81 First note det (A
Page 83 and 84: 3.3. THE MATHEMATICAL THEORY OF DET
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3.3. THE MATHEMATICAL THEORY OF DET
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3.4. THE CAYLEY HAMILTON THEOREM 97
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3.5. BLOCK MULTIPLICATION OF MATRIC
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3.5. BLOCK MULTIPLICATION OF MATRIC
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3.6. EXERCISES 103 derivatives so i
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Row Operations 4.1 Elementary Matri
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4.1. ELEMENTARY MATRICES 107 This h
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4.1. ELEMENTARY MATRICES 109 Now co
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4.2. THE RANK OF A MATRIX 111 So ho
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4.3. THE ROW REDUCED ECHELON FORM 1
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4.3. THE ROW REDUCED ECHELON FORM 1
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4.5. FREDHOLM ALTERNATIVE 117 Proof
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4.6. EXERCISES 119 5. ↑Consider t
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4.6. EXERCISES 121 18. Suppose A is
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Some Factorizations 5.1 LU Factoriz
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5.3. SOLVING LINEAR SYSTEMS USING A
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5.5. JUSTIFICATION FOR THE MULTIPLI
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5.6. EXISTENCE FOR THE PLU FACTORIZ
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5.7. THE QR FACTORIZATION 131 so it
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5.8. EXERCISES 133 Theorem 5.7.5 Le
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Linear Programming 6.1 Simple Geome
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6.2. THE SIMPLEX TABLEAU 137 set x
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6.2. THE SIMPLEX TABLEAU 139 lution
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6.3. THE SIMPLEX ALGORITHM 141 Let
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6.3. THE SIMPLEX ALGORITHM 143 the
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6.3. THE SIMPLEX ALGORITHM 145 by 2
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6.3. THE SIMPLEX ALGORITHM 147 I ne
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6.3. THE SIMPLEX ALGORITHM 149 Of c
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6.4. FINDING A BASIC FEASIBLE SOLUT
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6.5. DUALITY 153 Now recall that to
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6.5. DUALITY 155 and there are stil
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Spectral Theory Spectral Theory ref
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7.1. EIGENVALUES AND EIGENVECTORS O
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7.2. SOME APPLICATIONS OF EIGENVALU
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7.3. EXERCISES 167 Therefore, the e
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7.3. EXERCISES 169 22. Find the com
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7.3. EXERCISES 171 38. Let M be an
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7.4. SCHUR’S THEOREM 173 7.4 Schu
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7.4. SCHUR’S THEOREM 175 Proof: L
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7.4. SCHUR’S THEOREM 177 and furt
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7.4. SCHUR’S THEOREM 179 Proof: F
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7.6. QUADRATIC FORMS 181 7.6 Quadra
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7.7. SECOND DERIVATIVE TEST 183 Cor
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7.7. SECOND DERIVATIVE TEST 185 The
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7.9. ADVANCED THEOREMS 187 for find
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7.9. ADVANCED THEOREMS 189 Proof: D
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7.10. EXERCISES 191 Show that the a
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7.10. EXERCISES 193 27. Let f (x, y
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7.10. EXERCISES 195 40. In the abov
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7.10. EXERCISES 197 44. Show that i
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Vector Spaces And Fields 8.1 Vector
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8.2. SUBSPACES AND BASES 201 8.2.2
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8.2. SUBSPACES AND BASES 203 Defini
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8.3. LOTS OF FIELDS 205 such that 0
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8.3. LOTS OF FIELDS 207 and this is
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8.3. LOTS OF FIELDS 209 Lemma 8.3.9
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8.3. LOTS OF FIELDS 211 Definition
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8.3. LOTS OF FIELDS 213 Then, since
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8.3. LOTS OF FIELDS 215 Proposition
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8.3. LOTS OF FIELDS 217 where deg r
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8.4. EXERCISES 219 8.3.4 The Lindem
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8.4. EXERCISES 221 18. If you have
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8.4. EXERCISES 223 first is not har
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Linear Transformations 9.1 Matrix M
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9.3. THE MATRIX OF A LINEAR TRANSFO
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9.5. EXERCISES 243 9. ↑In the sit
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Linear Transformations Canonical Fo
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10.1. A THEOREM OF SYLVESTER, DIREC
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10.2. DIRECT SUMS, BLOCK DIAGONAL M
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10.3. CYCLIC SETS 251 while ⎛ ⎞
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10.3. CYCLIC SETS 253 Since β x is
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10.4. NILPOTENT TRANSFORMATIONS 255
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10.5. THE JORDAN CANONICAL FORM 257
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10.6. EXERCISES 263 8. Let ⎛ A =
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10.6. EXERCISES 265 Now use this in
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10.7. THE RATIONAL CANONICAL FORM 2
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10.8. UNIQUENESS 269 Thus the matri
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10.8. UNIQUENESS 271 The characteri
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10.9. EXERCISES 273 Then doing the
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Markov Chains And Migration Process
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11.1. REGULAR MARKOV MATRICES 277 O
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11.2. MIGRATION MATRICES 279 11.2 M
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11.3. MARKOV CHAINS 281 After many
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11.3. MARKOV CHAINS 283 and the fac
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11.4. EXERCISES 285 5. Show that wh
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Inner Product Spaces 12.1 General T
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12.2. THE GRAM SCHMIDT PROCESS 289
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12.2. THE GRAM SCHMIDT PROCESS 291
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12.3. RIESZ REPRESENTATION THEOREM
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12.4. THE TENSOR PRODUCT OF TWO VEC
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12.5. LEAST SQUARES 297 Lemma 12.5.
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12.7. EXERCISES 299 4. Let ||x||
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12.7. EXERCISES 301 (√ √ ) 2 C
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12.8. THE DETERMINANT AND VOLUME 30
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12.8. THE DETERMINANT AND VOLUME 30
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Self Adjoint Operators 13.1 Simulta
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13.1. SIMULTANEOUS DIAGONALIZATION
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13.2. SCHUR’S THEOREM 311 Proof:
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13.3. SPECTRAL THEORY OF SELF ADJOI
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13.3. SPECTRAL THEORY OF SELF ADJOI
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13.4. POSITIVE AND NEGATIVE LINEAR
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13.5. FRACTIONAL POWERS 319 Proof:
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13.5. FRACTIONAL POWERS 321 Proof:
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13.6. POLAR DECOMPOSITIONS 323 Proo
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13.7. AN APPLICATION TO STATISTICS
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13.8. THE SINGULAR VALUE DECOMPOSIT
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13.9. APPROXIMATION IN THE FROBENIU
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13.10. LEAST SQUARES AND SINGULAR V
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13.11. THE MOORE PENROSE INVERSE 33
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13.12. EXERCISES 335 8. Prove the C
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Norms For Finite Dimensional Vector
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339 This proves the first half of t
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341 Next consider the claim that ||
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14.1. THE P NORMS 343 14.1 The p No
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14.2. THE CONDITION NUMBER 345 Thus
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14.2. THE CONDITION NUMBER 347 Sinc
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14.3. THE SPECTRAL RADIUS 349 ⎛ =
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14.4. SERIES AND SEQUENCES OF LINEA
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14.4. SERIES AND SEQUENCES OF LINEA
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14.5. ITERATIVE METHODS FOR LINEAR
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14.6. THEORY OF CONVERGENCE 361 Lem
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14.7. EXERCISES 363 Proof: From Lem
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14.7. EXERCISES 365 11. Suppose ρ
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14.7. EXERCISES 367 Next take e tλ
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14.7. EXERCISES 369 24. Suppose A i
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Numerical Methods For Finding Eigen
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15.1. THE POWER METHOD FOR EIGENVAL
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15.2. THE QR ALGORITHM 387 each a v
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15.2. THE QR ALGORITHM 389 Proof: L
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15.2. THE QR ALGORITHM 391 where
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15.2. THE QR ALGORITHM 393 Corollar
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15.2. THE QR ALGORITHM 395 below th
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15.2. THE QR ALGORITHM 397 where R
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15.2. THE QR ALGORITHM 399 A matrix
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15.3. EXERCISES 401 where A k = Q k
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Positive Matrices Earlier theorems
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405 |μ| ≤λ 0 . But also, the fa
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407 which says x 0 −x 0 v T x 0 =
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409 must have the same argument for
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Functions Of Matrices The existence
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413 There is no convergence problem
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415 This gives us a nice definition
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Applications To Differential Equati
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C.3. LOCAL SOLUTIONS 419 C.3 Local
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C.4. FIRST ORDER LINEAR SYSTEMS 421
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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C.6. GENERAL GEOMETRIC THEORY 433 T
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C.7. THESTABLEMANIFOLD 435 Lemma C.
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C.7. THESTABLEMANIFOLD 437 ≤ e γ
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Compactness And Completeness D.0.1
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441 |a k − a l | < ε 2 .Thenifk>
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The Fundamental Theorem Of Algebra
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Fields And Field Extensions F.1 The
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F.2. THE FUNDAMENTAL THEOREM OF ALG
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F.2. THE FUNDAMENTAL THEOREM OF ALG
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F.3. TRANSCENDENTAL NUMBERS 451 F.3
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F.3. TRANSCENDENTAL NUMBERS 453 It
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F.3. TRANSCENDENTAL NUMBERS 455 Not
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F.3. TRANSCENDENTAL NUMBERS 457 Thi
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F.4. MORE ON ALGEBRAIC FIELD EXTENS
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F.5. THE GALOIS GROUP 465 a rationa
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F.5. THE GALOIS GROUP 467 Now let
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F.6. NORMAL SUBGROUPS 469 If H is a
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F.8. CONDITIONS FOR SEPARABILITY 47
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F.8. CONDITIONS FOR SEPARABILITY 47
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F.9. PERMUTATIONS 475 of f (x) ands
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F.9. PERMUTATIONS 477 smallest k
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F.10. SOLVABLE GROUPS 479 Then i 1
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F.10. SOLVABLE GROUPS 481 Thus ever
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F.11. SOLVABILITY BY RADICALS 483 A
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Bibliography [1] Apostol T., Calcul
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Answers To Selected Exercises G.1 E
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G.7. EXERCISES 489 15 Find the matr
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G.10. EXERCISES 491 11 It follows f
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G.13. EXERCISES 493 19 ⎛ ⎝ 4
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G.19. EXERCISES 495 8 λ 3 − λ 2
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G.23. EXERCISES 497 ⎧⎛ ⎞⎫
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INDEX 499 defective, 162 DeMoivre i
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INDEX 501 rotation, 235 linear tran
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INDEX 503 scalars, 18, 32, 37 Schur
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