Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

More documents

Recommendations

Info

70 ADVANCED ABSTRACT ALGEBRA Theorem 4 NM J s QP Jordan Forms If a square matrix A of order n has s linearly independent eigen vectors, then it is similar to a matrix J of the following form, called the Jordan Canonical form, LJ 1 M O J P 2 −1 J = Q AQ = Ο Ο in which each J i called a Jordan block, is a triangular matrix of the form J i = L NM λ i Mλ i 1 Ο Ο Ο Ο Ο Ο Ο 1 Ο λ i O QP where λ i is a single eigen value of A and s is the number of linearly independent eigen vectors of A. Remarks 1. If A has more than one linearly independent eigen vector, then same eigen value λ i may appear in several blocks. 2. If A has a full set of n linearly independent eigen vectors, then there have to be n Jordan blocks so that each Jordan block is just 1x1 matrix, and the corresponding Jordan canonical form is just the diagonal matrix with eigen values on the diagonal. Hence, a diagonal matrix is a particular case of the Jordan canonical form. The Jordan canonical form of a matrix can be completely determined by the multiplicites of the eigen values and the number of linearly independent eigen vectors in each of the eigen spaces. Definition Let V be an n-dimensional vector space over a field F. Two linear transformations S, T on V is said to similar if ∃ an invertible linear transformation C and V such that TA In terms of matrix form : Two n × n matrices A and B over F is said to be similar if ∃ an invertible n × n matrix C over F such that Proof of Theorem 4 is not important but its application is very important. (Interested readers may see proof in Herstein P. 301-303) Example 1 Let A be a 5 × 5 matrix with eigen value λ of multiplicity 5. Write all possible Jordan Canonical forms : We can get 7 Jordan canonical forms :
UNIT-III 71 1. J = only one linearly independent eigen vector belonging to Lλ NM 1 0 0 0 0 λ 1 0 0 0 0 λ 1 0 0 0 0 λ 1 0 0 0 0 λ O QP ∃ λλ J = L λ 0 NM 1 λ 1 0 λ λ 1 0 This Jordan canonical form consists of only one Jordan block with eigen value λ on the diagonal 2. two linearly independent eigen vectors belonging to . Then the Jordan canonical form of A is either one of the forms 0 λ 1 0 0 λ O QP J = , or Lλ NM λ 1 0 0 0 λ 1 0 0 0 λ 1 0 0 0 λ O QP Each of which consists of two Jordan blocks with eigen value λ on the diagonal. 3. ∃ three linearly independent eigen vectors belonging to Then the Jordan Canonical form of A is either one of the forms J = , or Lλ NM λ λ 1 0 0 λ 1 0 0 λ O QP 4. Each of which consists of three Jordan blocks with eigen value λ on the diagonal. four linearly independent eigen vectors belonging to . Then the Jordan canonical form of A is of the form 5. This consists of four Jordan blocks with eigen value λ on the diagonal. five linearly independent eigen vectors belonging to . Then the Jordan canonical form of A is of the form.
Page 1 and 2:
1 Advanced Abstract Algebra M.A./M.
Page 3 and 4:
Contents 3 Unit I 5 Unit II 49 Unit
Page 5 and 6:
UNIT-I 5 Unit-I Definition Group A
Page 7 and 8:
UNIT-I 7 2. In example 1, a 1 1 0 1
Page 9 and 10:
UNIT-I 9 Solutions: 1. CGbg≠ a φ
Page 11 and 12:
UNIT-I 11 ∴ Ha ⊆ H. To show let
Page 13 and 14:
UNIT-I 13 Corollary 4: (Feremat's L
Page 15 and 16:
UNIT-I 15 Definition: Group Homomor
Page 17 and 18:
UNIT-I 17 Proof: Consider the diagr
Page 19 and 20:
UNIT-I 19 ( G ) N G ( H ) ≅ H N D
Page 21 and 22:
UNIT-1 21 Theorem 8. Let H be a sub
Page 23 and 24:
UNIT-1 23 From (i) and (ii), we hav
Page 25 and 26:
UNIT-I 25 1 Note that αβ = 2 2 1
Page 27 and 28:
UNIT-1 27 In this case the length o
Page 29 and 30:
UNIT-1 29 | n Theorem 18. The set o
Page 31 and 32:
UNIT-I b = φ ( φ φ ) ( ) g h t x
Page 33 and 34:
UNIT-I 33 This is impossible. So no
Page 35 and 36:
UNIT-I 35 integers x and y. Note th
Page 37 and 38:
UNIT-I 37 O(7) = 4, as 7 1 = 7, 7 2
Page 39 and 40:
UNIT-I 39 Put a x mt , = b = x Then
Page 41 and 42:
UNIT-I n iii. N = ∈Gs, show that
Page 43 and 44:
UNIT-I 43 abab------ab = a n b n
Page 45 and 46:
UNIT-I Q.20. Q.21. Q.22. Q.23. Q.24
Page 47 and 48:
UNIT-I 47 Remark : If G is abelian,
Page 49 and 50:
UNIT-IV 81 Definition. Let S be a s
Page 51 and 52: UNIT-IV 83 (r+s) − (r 1 +s 1 ) =
Page 53 and 54: UNIT-IV 85 and = ψ(r+K) + ψ(s+K)
Page 55 and 56: UNIT-IV 87 M ⊂ M′ ⊂ R M′ =
Page 57 and 58: UNIT-IV 89 We define addition and m
Page 59 and 60: UNIT-IV 91 be the set of the ordere
Page 61 and 62: UNIT-IV 93 (iii) a + b − a b
Page 63 and 64: UNIT-IV 95 and Z/A ~ Im(f) = f (z)
Page 65 and 66: UNIT-IV 97 If a n = 0 for all n ≥
Page 67 and 68: UNIT-IV 99 x 3 = (0, 0, 0, 1,…) .
Page 69 and 70: UNIT-IV 101 Then either (i) or n i
Page 71 and 72: UNIT-IV 103 deg f(x) = 0 and deg g(
Page 73 and 74: UNIT-IV 105 If x, y ∈ A , then th
Page 75 and 76: UNIT-IV 107 The possibility φ(r) <
Page 77 and 78: UNIT-IV 109 Lemma 4. If f(x) is an
Page 79 and 80: UNIT-IV 111 Proof. Let, if possible
Page 81 and 82: UNIT-II 49 Unit-II Definition: Comp
Page 83 and 84: UNIT-II 51 If r = 1, then G is simp
Page 85 and 86: UNIT-II 53 group of the Commutator
Page 87 and 88: UNIT-II di + 1 idi + 1 i di + 1 idi
Page 89 and 90: UNIT-II 57 we write the refinement
Page 91 and 92: UNIT-II 59 of length e in which eac
Page 93 and 94: UNIT-II 61 of nilpotency of G) , so
Page 95 and 96: UNIT-III 63 n.v = R i.e. abelian gr
Page 97 and 98: UNIT-III 65 Theorem 1 Let M, N be R
Page 99 and 100: UNIT-III 67 F ( BA ad 1'= = , 1a d(
Page 101: UNIT-III 69 Vector space over F. Le
Page 105 and 106: UNIT-III Hence J = −L NM 4 0 0 0
Page 107 and 108: UNIT-III 75 Example 5 The direct su
Page 109 and 110: UNIT-IV 77 A. B ≠ B. A. If IR, th
Page 111 and 112: UNIT-IV 79 Example: (D, +, .) is a
Page 113 and 114: UNIT-IV 81 Examples 1. The polynomi
Page 115 and 116: UNIT-V 129 (Characteristic of a rin
Page 117 and 118: UNIT-V 131 As F = p dΘ F ≅ Z p i
Page 119 and 120: UNIT-V 133 In this case is called t
Page 121 and 122: + c c+ 0 0 1 c c+ 1 1 1 0 1+ c c c
Page 123 and 124: UNIT-V 137 element of defines a per
show all

Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?