A Proof of Fundamental Theorem of Algebra Through Linear Algebra due to Derksen. Anant R. Shastri. February 13, 2011

Transcription

1 A Proof of Fundamental Theorem of Algebra Through Linear Algebra due to Derksen February 13, 2011

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3 MA106-Linear Algebra 2011

4 MA106-Linear Algebra 2011 We present a proof of the Fundamental Theorem of Algebra (FTA) Every non constant polynomial in one variable with coefficients in C has a root in C.

5 MA106-Linear Algebra 2011 We present a proof of the Fundamental Theorem of Algebra (FTA) Every non constant polynomial in one variable with coefficients in C has a root in C. through a sequence of easily do-able exercises. The proof uses elementary linear algebra except that we begin with the intermediate value theorem in proving that every odd degee real polynomial has a real root.

6 Based on an article in

7 Based on an article in Amer. Math. Monthly- 110,(2003), pp

8 Sketch of the Proof In what follows K will denote any field. However, we need worry about only two cases here: K is either R or C.

9 We begin with a basic result in real analysis:

10 We begin with a basic result in real analysis: Intermediate Value Theorem

11 We begin with a basic result in real analysis: Intermediate Value Theorem which immediately yields:

12 We begin with a basic result in real analysis: Intermediate Value Theorem which immediately yields: Every odd degree polynomial p(t) R[t] has a real root. It should be noted that there is no purely algebraic proof of FTA which does not use IVT. Indeed, all proofs of FTA use IVT explicitly or implicitly. The simple reason is that IVT is equivalent to any other axiom that is used in the construction of real numbers.

13 We begin with a basic result in real analysis: Intermediate Value Theorem which immediately yields: Every odd degree polynomial p(t) R[t] has a real root. It should be noted that there is no purely algebraic proof of FTA which does not use IVT. Indeed, all proofs of FTA use IVT explicitly or implicitly. The simple reason is that IVT is equivalent to any other axiom that is used in the construction of real numbers. From now onwards we only use linear algebra.

14 Sketch of the Proof Companion Matrix Let p(t) = t n + a 1 t n a n be a monic polynomial of degree n. Its companion matrix C p is defined to be the n n matrix C p = a n a n 1 a 1.

15 Sketch of the Proof Companion Matrix Let p(t) = t n + a 1 t n a n be a monic polynomial of degree n. Its companion matrix C p is defined to be the n n matrix C p = a n a n 1 a 1. Ex.1. Show that the characteristic polynomial of C p is p, i.e., det(c p ti n ) = ( 1) n p(t).

16 Ex. 2. Show that every non constant polynomial p(t) K[t] of degree n has a root in K iff every endomorphism K n K n has an eigenvalue in K

17 Ex. 2. Show that every non constant polynomial p(t) K[t] of degree n has a root in K iff every endomorphism K n K n has an eigenvalue in K iff every n n matrix over K has an eigenvalue in K.

18 Ex. 2. Show that every non constant polynomial p(t) K[t] of degree n has a root in K iff every endomorphism K n K n has an eigenvalue in K iff every n n matrix over K has an eigenvalue in K. Recall that by an endomorphism of a vector space V we mean a linear transformation V V.

19 Ex. 2. Show that every non constant polynomial p(t) K[t] of degree n has a root in K iff every endomorphism K n K n has an eigenvalue in K iff every n n matrix over K has an eigenvalue in K. Recall that by an endomorphism of a vector space V we mean a linear transformation V V. Ex. 3 Show that every R-linear map f : R 2n+1 R 2n+1 has a real eigenvalue.

20 Sketch of the Proof For positive integers r, let S 1 (K, r) denote the following statement: Every endomorphism A : K n K n has an eigenvector for all n not divisible by 2 r. Also let S 2 (K, r) denote the statement Any two commuting endomorphisms K n K n have a common eigenvector for all n not divisible by 2 r.

21 Sketch of the Proof For positive integers r, let S 1 (K, r) denote the following statement: Every endomorphism A : K n K n has an eigenvector for all n not divisible by 2 r. Also let S 2 (K, r) denote the statement Any two commuting endomorphisms K n K n have a common eigenvector for all n not divisible by 2 r. We shall prove one result over any field K and two special results for K = R and C.

22 Sketch of the Proof For positive integers r, let S 1 (K, r) denote the following statement: Every endomorphism A : K n K n has an eigenvector for all n not divisible by 2 r. Also let S 2 (K, r) denote the statement Any two commuting endomorphisms K n K n have a common eigenvector for all n not divisible by 2 r. We shall prove one result over any field K and two special results for K = R and C.

23 Sketch of Proof (a) S 1 (K, r) = S 2 (K, r), r 1.

24 Sketch of Proof (a) S 1 (K, r) = S 2 (K, r), r 1. (b) S 2 (R, 1) = S 1 (C, 1).

25 Sketch of Proof (a) S 1 (K, r) = S 2 (K, r), r 1. (b) S 2 (R, 1) = S 1 (C, 1). (c) S 1 (C, r) = S 1 (C, r + 1), r 1.

26 Sketch of Proof (a) S 1 (K, r) = S 2 (K, r), r 1. (b) S 2 (R, 1) = S 1 (C, 1). (c) S 1 (C, r) = S 1 (C, r + 1), r 1. Begin with IVT which is the same as S 1 (R, 1). Putting K = R in (a) we get S 2 (R, 1).Now (b) gives S 1 (C, 1) and repeated application of (c) gives S 1 (C, k) for all k 1.

27 Sketch of Proof (a) S 1 (K, r) = S 2 (K, r), r 1. (b) S 2 (R, 1) = S 1 (C, 1). (c) S 1 (C, r) = S 1 (C, r + 1), r 1. Begin with IVT which is the same as S 1 (R, 1). Putting K = R in (a) we get S 2 (R, 1).Now (b) gives S 1 (C, 1) and repeated application of (c) gives S 1 (C, k) for all k 1. Given n, write n = 2 k l where l is odd. Then S 1 (C, k + 1) implies that every polynomial of degree n has a root. And that completes the proof of FTA.

28 Ex. 4. Prove statement (a): S 1 (K, r) = S 2 (K, r).

29 Ex. 4. Prove statement (a): S 1 (K, r) = S 2 (K, r). We shall prove this by induction on the dimension n of the vector space.

30 Ex. 4. Prove statement (a): S 1 (K, r) = S 2 (K, r). We shall prove this by induction on the dimension n of the vector space. For n = 1 this is clear, since any non zero vector is an eigenvector for all endomorphisms.

31 Ex. 4. Prove statement (a): S 1 (K, r) = S 2 (K, r). We shall prove this by induction on the dimension n of the vector space. For n = 1 this is clear, since any non zero vector is an eigenvector for all endomorphisms. So assume that the statement is true for smaller values of n and let A, B be two commuting n n matrices over K, and n is not divisible by 2 r.

32 S 1 (K, r) implies S 2 (K, r) Let λ be an eigenvalue of A and put V 1 = N (A λ I ), V 2 = R(A λ I ). Since B commutes with A, it follows that B commutes with A λi ) also. Therefore B restricts to endomorphisms β : V 1 V 1 and β : V 2 V 2. (Refer: Tut. sheet 9.18.)

33 S 1 (K, r) implies S 2 (K, r) Let λ be an eigenvalue of A and put V 1 = N (A λ I ), V 2 = R(A λ I ). Since B commutes with A, it follows that B commutes with A λi ) also. Therefore B restricts to endomorphisms β : V 1 V 1 and β : V 2 V 2. (Refer: Tut. sheet 9.18.) By rank-nullity theorem, dim V 1 + dim V 2 = n and hence dim V 1 or dim V 2 is not divisible by 2 r.

34 S 1 (K, r) implies S 2 (K, r) Let λ be an eigenvalue of A and put V 1 = N (A λ I ), V 2 = R(A λ I ). Since B commutes with A, it follows that B commutes with A λi ) also. Therefore B restricts to endomorphisms β : V 1 V 1 and β : V 2 V 2. (Refer: Tut. sheet 9.18.) By rank-nullity theorem, dim V 1 + dim V 2 = n and hence dim V 1 or dim V 2 is not divisible by 2 r. If dim V 1 is not divisible by 2 r then β : V 1 V 1 has an eigenvector which is also the eigenvector for A.

35 S 1 (K, r) implies S 2 (K, r) Let λ be an eigenvalue of A and put V 1 = N (A λ I ), V 2 = R(A λ I ). Since B commutes with A, it follows that B commutes with A λi ) also. Therefore B restricts to endomorphisms β : V 1 V 1 and β : V 2 V 2. (Refer: Tut. sheet 9.18.) By rank-nullity theorem, dim V 1 + dim V 2 = n and hence dim V 1 or dim V 2 is not divisible by 2 r. If dim V 1 is not divisible by 2 r then β : V 1 V 1 has an eigenvector which is also the eigenvector for A. If not, then dim V 2 < n and not divisible by 2 r. Now we consider α, β : V 2 V 2 which are restrictions of A and B and hence are commuting endomorphisms. By induction we are through there is a common eigen vector v V 2 V for α and β which is then a common eigen vector for A and B as well.

36 Ex. 5 Show that the space HERM n (C) of all complex Hermitian n n matrices is a R vector space of dimension n 2.

37 Ex. 5 Show that the space HERM n (C) of all complex Hermitian n n matrices is a R vector space of dimension n 2. (Refer to Exercise 8.5 and a question in the Quiz.)

38 Ex. 5 Show that the space HERM n (C) of all complex Hermitian n n matrices is a R vector space of dimension n 2. (Refer to Exercise 8.5 and a question in the Quiz.) Recall that a matrix A is hermitian if it is equal to its transpose conjugate A. and the next one is from 8.6. Ex. 6 Given A M n (C), the mappings α A (B) = 1 2 (AB +BA ); β A (B) = 1 2ı (AB BA ) define R-linear maps HERM n (C) HERM n (C). Show that α A, β A commute with each other.

39 Answer: 4ıα A β A (B) = 2α A (AB BA ) = [A(AB BA ) + (AB BA )A ] = [A(AB + BA ) (AB + BA )A ] = 2ıβ A (AB + BA ) = 4ıβ A α A (B).

40 Ex. 7. If α A and β A have a common eigenvector then A has an eigenvalue in C.

41 Ex. 7. If α A and β A have a common eigenvector then A has an eigenvalue in C. Answer: If α A (B) = λb and β A (B) = µb then consider AB = (α A + ıβ A )(B) = (λ + ıµ)b.

42 Ex. 7. If α A and β A have a common eigenvector then A has an eigenvalue in C. Answer: If α A (B) = λb and β A (B) = µb then consider AB = (α A + ıβ A )(B) = (λ + ıµ)b. Since B is an eigenvector, at least one of the column vectors of B, say u 0.

43 Ex. 7. If α A and β A have a common eigenvector then A has an eigenvalue in C. Answer: If α A (B) = λb and β A (B) = µb then consider AB = (α A + ıβ A )(B) = (λ + ıµ)b. Since B is an eigenvector, at least one of the column vectors of B, say u 0. It follows that Au = (λ + ıµ)u and hence λ + ıµ is an eigenvalue for A.

44 Ex.8. Prove statement (b): S 2 (R, 1) = S 1 (C, 1). (Exercise 11.4)

45 Ex.8. Prove statement (b): S 2 (R, 1) = S 1 (C, 1). (Exercise 11.4) Solution: By Ex.6, given A M n (C), α A, β A : Herm n (C) Herm n (C) are two commuting endomorphisms of the real vector space Herm n which is of dimension n 2, by ex. 5.

46 Ex.8. Prove statement (b): S 2 (R, 1) = S 1 (C, 1). (Exercise 11.4) Solution: By Ex.6, given A M n (C), α A, β A : Herm n (C) Herm n (C) are two commuting endomorphisms of the real vector space Herm n which is of dimension n 2, by ex. 5. If n is odd, so is n 2.

47 Ex.8. Prove statement (b): S 2 (R, 1) = S 1 (C, 1). (Exercise 11.4) Solution: By Ex.6, given A M n (C), α A, β A : Herm n (C) Herm n (C) are two commuting endomorphisms of the real vector space Herm n which is of dimension n 2, by ex. 5. If n is odd, so is n 2. By Ex.4, it follows that α A, β A have a common eigenvector.

48 Ex.8. Prove statement (b): S 2 (R, 1) = S 1 (C, 1). (Exercise 11.4) Solution: By Ex.6, given A M n (C), α A, β A : Herm n (C) Herm n (C) are two commuting endomorphisms of the real vector space Herm n which is of dimension n 2, by ex. 5. If n is odd, so is n 2. By Ex.4, it follows that α A, β A have a common eigenvector. By Ex. 7, A has an eigenvalue.

49 Ex. 9. Show that the space Sym n (K) of symmetric n n matrices forms a subspace of dimension n(n + 1)/2 of M n (K). (Exercise 8.5)

50 Ex. 9. Show that the space Sym n (K) of symmetric n n matrices forms a subspace of dimension n(n + 1)/2 of M n (K). (Exercise 8.5) The next one is Ex

51 Ex. 9. Show that the space Sym n (K) of symmetric n n matrices forms a subspace of dimension n(n + 1)/2 of M n (K). (Exercise 8.5) The next one is Ex Ex. 10 Given A M n (K), show that φ A : B 1 2 (AB + BAt ); ψ A : B ABA t define two commuting endomorphisms of Sym n (K). Show that if B is a common eigenvector of φ A, ψ A, then (A 2 + aa + b I n )B = 0 for some a, b K; further if K = C, conclude that A has an eigenvalue.

52 Answer: The first part is easy.

53 Answer: The first part is easy. To see the second part, suppose and φ A (B) = AB + BA t = λb ψ A (B) = ABA t = µb

54 Answer: The first part is easy. To see the second part, suppose and φ A (B) = AB + BA t = λb ψ A (B) = ABA t = µb Multiply first relation by A on the left and use the second to obtain which is the same as A 2 B + µb λab = 0 (A 2 λa + µ I n )B = 0.

55 For the last part, first observe that since B is an eigenvector there is at least one column vector v which is non zero. Therefore (A 2 + aa + b I n )v = 0.

56 For the last part, first observe that since B is an eigenvector there is at least one column vector v which is non zero. Therefore (A 2 + aa + b I n )v = 0. Now write A 2 + aa + bi n = (A λ 1 I n )(A λ 2 I n ). If (A λ 2 I n )v = 0, then λ 2 is an eigenvalue of A and we are through.

57 For the last part, first observe that since B is an eigenvector there is at least one column vector v which is non zero. Therefore (A 2 + aa + b I n )v = 0. Now write A 2 + aa + bi n = (A λ 1 I n )(A λ 2 I n ). If (A λ 2 I n )v = 0, then λ 2 is an eigenvalue of A and we are through. Otherwise put u = (A λ 2 I n )v 0. Then (A λ 1 I n )u = 0 and hence λ 1 is an eigenvalue of A.

58 Ex. 11 Prove statement (c) S 1 (C, r) = S 1 (C, r + 1). Hence conclude that S 1 (C, r) is true for all r 1.

59 Ex. 11 Prove statement (c) S 1 (C, r) = S 1 (C, r + 1). Hence conclude that S 1 (C, r) is true for all r 1. Answer: Let n = 2 k l where 1 k r and l is odd. Let A M n (C). Then φ A, ψ A are two mutually commuting operators on Sym n (C) which is of dimension n(n + 1)/2 = 2 k 1 l(2 k l + 1) which is not divisible by 2 r.

60 Ex. 11 Prove statement (c) S 1 (C, r) = S 1 (C, r + 1). Hence conclude that S 1 (C, r) is true for all r 1. Answer: Let n = 2 k l where 1 k r and l is odd. Let A M n (C). Then φ A, ψ A are two mutually commuting operators on Sym n (C) which is of dimension n(n + 1)/2 = 2 k 1 l(2 k l + 1) which is not divisible by 2 r. By Ex. 10, with K = C along with the induction hypothesis, it follows that A has an eigenvalue.

61 We summerise what we did.

62 We summerise what we did. In order to prove that every non constant polynomial over complex numbers has a root we note that

63 We summerise what we did. In order to prove that every non constant polynomial over complex numbers has a root we note that by Ex.2, it is enough to show that every n n complex matrix has an eigenvalue.

64 We summerise what we did. In order to prove that every non constant polynomial over complex numbers has a root we note that by Ex.2, it is enough to show that every n n complex matrix has an eigenvalue. By Ex.8, this holds for odd n. This implies that S 1 (C, 1). By Ex. 11 applied repeatedly, we get S 1 (C, r) for all r. Given any polynomial of degree n, write n = 2 r l, where l is odd. Then S 1 (C, r + 1) implies that every polynomial of degree n has a root.

65 THANK YOU FOR YOUR ATTENTION