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![Page 1: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/1.jpg)
FUNDAMENTAL THEOREM of ALGEBRAthrough Linear Algebra
Anant R. ShastriI. I. T. Bombay
18th August 2012
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 2: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/2.jpg)
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 3: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/3.jpg)
I Teachers’ Enrichment Programme inLinear Algebra at IITB, Aug. 2012
I We present a proof ofFundamental Theorem of Algebrathrough a sequence of easily do-able exercises inLinear Algebra except one single result inelementary real anaylsis, viz.,the intermediate value theorem.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 4: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/4.jpg)
I Teachers’ Enrichment Programme inLinear Algebra at IITB, Aug. 2012
I We present a proof ofFundamental Theorem of Algebrathrough a sequence of easily do-able exercises inLinear Algebra except one single result inelementary real anaylsis, viz.,the intermediate value theorem.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 5: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/5.jpg)
I Based on an article in
I Amer. Math. Monthly- 110,(2003), pp.620-623. by Derksen, a GermanMathematician.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 6: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/6.jpg)
I Based on an article in
I Amer. Math. Monthly- 110,(2003), pp.620-623. by Derksen, a GermanMathematician.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 7: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/7.jpg)
I In what follows K will denote any field.However, we need to take K to be either R or C.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 8: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/8.jpg)
I Ex. 1 Show that every odd degree polynomialP(t) ∈ R[t] has a real root.
I This is where Intermediate Value Theorem isused.
I We may assume P(t) = tn + a1tn−1 + · · · and
see that as t → +∞, p(t)→ +∞, and
I As t → −∞,P(t)→ −∞. It follows that thereexist t0 ∈ R such that P(t0) = 0.
I From now onwards we only use linear algebra.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 9: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/9.jpg)
I Ex. 1 Show that every odd degree polynomialP(t) ∈ R[t] has a real root.
I This is where Intermediate Value Theorem isused.
I We may assume P(t) = tn + a1tn−1 + · · · and
see that as t → +∞, p(t)→ +∞, and
I As t → −∞,P(t)→ −∞. It follows that thereexist t0 ∈ R such that P(t0) = 0.
I From now onwards we only use linear algebra.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 10: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/10.jpg)
I Ex. 1 Show that every odd degree polynomialP(t) ∈ R[t] has a real root.
I This is where Intermediate Value Theorem isused.
I We may assume P(t) = tn + a1tn−1 + · · · and
see that as t → +∞, p(t)→ +∞, and
I As t → −∞,P(t)→ −∞. It follows that thereexist t0 ∈ R such that P(t0) = 0.
I From now onwards we only use linear algebra.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 11: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/11.jpg)
I Ex. 1 Show that every odd degree polynomialP(t) ∈ R[t] has a real root.
I This is where Intermediate Value Theorem isused.
I We may assume P(t) = tn + a1tn−1 + · · · and
see that as t → +∞, p(t)→ +∞, and
I As t → −∞,P(t)→ −∞. It follows that thereexist t0 ∈ R such that P(t0) = 0.
I From now onwards we only use linear algebra.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 12: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/12.jpg)
I Ex. 1 Show that every odd degree polynomialP(t) ∈ R[t] has a real root.
I This is where Intermediate Value Theorem isused.
I We may assume P(t) = tn + a1tn−1 + · · · and
see that as t → +∞, p(t)→ +∞, and
I As t → −∞,P(t)→ −∞. It follows that thereexist t0 ∈ R such that P(t0) = 0.
I From now onwards we only use linear algebra.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 13: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/13.jpg)
I Companion Matrix LetP(t) = tn + a1t
n−1 + · · ·+ an be a monicpolynominal of degree n. Its companian matrixCP is defined to be the n × n matrix
CP =
0 1 0 · · · 0 00 0 1 · · · 0 0...
......
...0 · · · · · · 0 1−an −an−1 · · · · · · −a1
.
I Ex. 2 Show that det(tIn − CP) = P(t).
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 14: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/14.jpg)
I Companion Matrix LetP(t) = tn + a1t
n−1 + · · ·+ an be a monicpolynominal of degree n. Its companian matrixCP is defined to be the n × n matrix
CP =
0 1 0 · · · 0 00 0 1 · · · 0 0...
......
...0 · · · · · · 0 1−an −an−1 · · · · · · −a1
.I Ex. 2 Show that det(tIn − CP) = P(t).
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 15: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/15.jpg)
Ex. 3 Show that every non constant polynomialP(t) ∈ K[t] of degree n has a root in K iff everylinear map Kn → Kn has an eigen value in K.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 16: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/16.jpg)
I Ex. 4 Show that every R-linear mapf : R2n+1 → R2n+1 has real eigen value.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 17: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/17.jpg)
I Ex. 5 Show that the space HERMn(C) of allcomplex Hermitian n × n matrices (i.e., all Asuch that A∗ = A) is a R-vector space ofdimension n2.
I Ex. 6 Given A ∈ Mn(C), the mappings
αA(B) =1
2(AB+BA∗); βA(B) =
1
2ı(AB−BA∗)
define R-linear maps HERMn(C)→ HERMn(C).Show that αA, βA commute with each other.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 18: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/18.jpg)
I Ex. 5 Show that the space HERMn(C) of allcomplex Hermitian n × n matrices (i.e., all Asuch that A∗ = A) is a R-vector space ofdimension n2.
I Ex. 6 Given A ∈ Mn(C), the mappings
αA(B) =1
2(AB+BA∗); βA(B) =
1
2ı(AB−BA∗)
define R-linear maps HERMn(C)→ HERMn(C).Show that αA, βA commute with each other.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 19: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/19.jpg)
I Answer:
αA ◦ βA(B) = 14ıαA(AB − BA∗)
= 14ı[A(AB − BA∗) + (AB − BA∗)A∗]
= 14ı[A(AB + BA∗)− (AB + BA∗)A∗]
= 14ıβA ◦ αA(B).
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 20: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/20.jpg)
I Ex. 7 If αA and βA have a common eigen vectorthen A has an eigen value in C.
I Answer: If αA(B) = λB and βA(B) = µB thenconsider
AB = (αA + ıβA)(B) = (λ + ıµ)B .
I Since B is an eigen vector, at least one of thecolumn vectors , say u 6= 0.
I It follows that Au = (λ+ ıµ)u and hence λ+ ıµis an eigen value for A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 21: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/21.jpg)
I Ex. 7 If αA and βA have a common eigen vectorthen A has an eigen value in C.
I Answer: If αA(B) = λB and βA(B) = µB thenconsider
AB = (αA + ıβA)(B) = (λ + ıµ)B .
I Since B is an eigen vector, at least one of thecolumn vectors , say u 6= 0.
I It follows that Au = (λ+ ıµ)u and hence λ+ ıµis an eigen value for A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 22: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/22.jpg)
I Ex. 7 If αA and βA have a common eigen vectorthen A has an eigen value in C.
I Answer: If αA(B) = λB and βA(B) = µB thenconsider
AB = (αA + ıβA)(B) = (λ + ıµ)B .
I Since B is an eigen vector, at least one of thecolumn vectors , say u 6= 0.
I It follows that Au = (λ+ ıµ)u and hence λ+ ıµis an eigen value for A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 23: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/23.jpg)
I Ex. 7 If αA and βA have a common eigen vectorthen A has an eigen value in C.
I Answer: If αA(B) = λB and βA(B) = µB thenconsider
AB = (αA + ıβA)(B) = (λ + ıµ)B .
I Since B is an eigen vector, at least one of thecolumn vectors , say u 6= 0.
I It follows that Au = (λ+ ıµ)u and hence λ+ ıµis an eigen value for A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 24: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/24.jpg)
I Ex. 8 Show that any two commuting linearmaps α, β : R2n+1 → R2n+1 have a commoneigen vector.
I Answer: Use induction and subspaces kernel andimage of α− λIn where λ is an eigen value of α.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 25: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/25.jpg)
I Ex. 8 Show that any two commuting linearmaps α, β : R2n+1 → R2n+1 have a commoneigen vector.
I Answer: Use induction and subspaces kernel andimage of α− λIn where λ is an eigen value of α.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 26: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/26.jpg)
I Put Ker α− λIn = V , andRange(α− λIn) = W .
I Then β(V ) ⊂ V and β(W ) ⊂ W . Alsoα(V ) ⊂ V ;α(W ) ⊂ W .
I If V is of odd dimension then any eigen vectorof β will do.
I Otherwise W is odd dimension strictly less than2n + 1. So induction works for α, β : W → W .
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 27: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/27.jpg)
I Put Ker α− λIn = V , andRange(α− λIn) = W .
I Then β(V ) ⊂ V and β(W ) ⊂ W . Alsoα(V ) ⊂ V ;α(W ) ⊂ W .
I If V is of odd dimension then any eigen vectorof β will do.
I Otherwise W is odd dimension strictly less than2n + 1. So induction works for α, β : W → W .
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 28: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/28.jpg)
I Put Ker α− λIn = V , andRange(α− λIn) = W .
I Then β(V ) ⊂ V and β(W ) ⊂ W . Alsoα(V ) ⊂ V ;α(W ) ⊂ W .
I If V is of odd dimension then any eigen vectorof β will do.
I Otherwise W is odd dimension strictly less than2n + 1. So induction works for α, β : W → W .
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 29: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/29.jpg)
I Put Ker α− λIn = V , andRange(α− λIn) = W .
I Then β(V ) ⊂ V and β(W ) ⊂ W . Alsoα(V ) ⊂ V ;α(W ) ⊂ W .
I If V is of odd dimension then any eigen vectorof β will do.
I Otherwise W is odd dimension strictly less than2n + 1. So induction works for α, β : W → W .
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 30: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/30.jpg)
I Ex. 9 Every C-linear map A : C2n+1 → C2n+1 hasan eigen value.
I Solution: By Ex. 5 and Ex. 8,αA, βA : Herm2n+1 → Herm2n+1 have a commoneigenvector.
I By Ex. 7, A has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 31: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/31.jpg)
I Ex. 9 Every C-linear map A : C2n+1 → C2n+1 hasan eigen value.
I Solution: By Ex. 5 and Ex. 8,αA, βA : Herm2n+1 → Herm2n+1 have a commoneigenvector.
I By Ex. 7, A has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 32: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/32.jpg)
I Ex. 9 Every C-linear map A : C2n+1 → C2n+1 hasan eigen value.
I Solution: By Ex. 5 and Ex. 8,αA, βA : Herm2n+1 → Herm2n+1 have a commoneigenvector.
I By Ex. 7, A has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 33: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/33.jpg)
I Ex. 10 Show that the space Symn(K) ofsymmetric n × n matrices forms a subspace ofdimension n(n + 1)/2 of Mn(K).
I Ex. 11 Given A ∈ Mn(K), show that
φA : B 7→ 1
2(AB + BAt); ψA : B 7→ ABAt
define two commuting endomorphisms ofSymn(K). Show that if B is a common eigenvector of φA, ψA then (A2 + aA + bId)B = 0 forsome a, b ∈ K. Further if K = C, conclude thatA has an eigen value.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 34: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/34.jpg)
I Ex. 10 Show that the space Symn(K) ofsymmetric n × n matrices forms a subspace ofdimension n(n + 1)/2 of Mn(K).
I Ex. 11 Given A ∈ Mn(K), show that
φA : B 7→ 1
2(AB + BAt); ψA : B 7→ ABAt
define two commuting endomorphisms ofSymn(K). Show that if B is a common eigenvector of φA, ψA then (A2 + aA + bId)B = 0 forsome a, b ∈ K. Further if K = C, conclude thatA has an eigen value.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 35: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/35.jpg)
I Answer: The first part is easy.
I To see the second part, suppose
φA(B) = AB + BAt = λB
andψA(B) = ABAt = µB
I Multiply first relation by A on the left and usethe second to obtain
A2B + µB − λAB = 0
which is the same as
(A2 − λA + µId)B = 0.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 36: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/36.jpg)
I Answer: The first part is easy.I To see the second part, suppose
φA(B) = AB + BAt = λB
andψA(B) = ABAt = µB
I Multiply first relation by A on the left and usethe second to obtain
A2B + µB − λAB = 0
which is the same as
(A2 − λA + µId)B = 0.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 37: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/37.jpg)
I Answer: The first part is easy.I To see the second part, suppose
φA(B) = AB + BAt = λB
andψA(B) = ABAt = µB
I Multiply first relation by A on the left and usethe second to obtain
A2B + µB − λAB = 0
which is the same as
(A2 − λA + µId)B = 0.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 38: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/38.jpg)
I For the last part, first observe that since B is aneigen vector there is atleast one column vector vwhich is non zero. Therefore(A2 + aA + b)v = 0.
I Now writeA2 + aA + bId = (A− λ1Id)(A− λ2Id). If(A− λ2Id)v = 0 then λ2 is an eigen value of A.and we are through.
I Otherwise u = (A− λ2Id) 6= 0 and(A− λ1Id)u = 0 and hence λ1 is an eigen valueof A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 39: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/39.jpg)
I For the last part, first observe that since B is aneigen vector there is atleast one column vector vwhich is non zero. Therefore(A2 + aA + b)v = 0.
I Now writeA2 + aA + bId = (A− λ1Id)(A− λ2Id). If(A− λ2Id)v = 0 then λ2 is an eigen value of A.and we are through.
I Otherwise u = (A− λ2Id) 6= 0 and(A− λ1Id)u = 0 and hence λ1 is an eigen valueof A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 40: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/40.jpg)
I For the last part, first observe that since B is aneigen vector there is atleast one column vector vwhich is non zero. Therefore(A2 + aA + b)v = 0.
I Now writeA2 + aA + bId = (A− λ1Id)(A− λ2Id). If(A− λ2Id)v = 0 then λ2 is an eigen value of A.and we are through.
I Otherwise u = (A− λ2Id) 6= 0 and(A− λ1Id)u = 0 and hence λ1 is an eigen valueof A.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 41: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/41.jpg)
I Let S1(K, r) denote the following statement:Any endomorphism A : Kn → Kn has an eigenvalue for all n not divisible by 2r . Let S2(K, r)denote the statement: Any two commutingendomorphisms A1,A2 : Kn → Kn have acommon eigen vector for all n not divisible by 2r .
I Ex. 12 Prove that S1(K, r) =⇒ S2(K, r).
I Answer: Exactly as in Ex. 8.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 42: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/42.jpg)
I Let S1(K, r) denote the following statement:Any endomorphism A : Kn → Kn has an eigenvalue for all n not divisible by 2r . Let S2(K, r)denote the statement: Any two commutingendomorphisms A1,A2 : Kn → Kn have acommon eigen vector for all n not divisible by 2r .
I Ex. 12 Prove that S1(K, r) =⇒ S2(K, r).
I Answer: Exactly as in Ex. 8.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 43: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/43.jpg)
I Let S1(K, r) denote the following statement:Any endomorphism A : Kn → Kn has an eigenvalue for all n not divisible by 2r . Let S2(K, r)denote the statement: Any two commutingendomorphisms A1,A2 : Kn → Kn have acommon eigen vector for all n not divisible by 2r .
I Ex. 12 Prove that S1(K, r) =⇒ S2(K, r).
I Answer: Exactly as in Ex. 8.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 44: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/44.jpg)
I Ex. 13 Prove that S1(C, r) =⇒ S1(C, r + 1).
I Answer:Let n = 2km where m is odd and k ≤ r . LetA ∈ Mn(C). Then φA, ψA are two mutuallycommuting operators on Symn(C) which is ofdimension n(n + 1)/2 = 2k−1m(2km + 1) whichis not divisible by 2r .
I Therefore S1(C, r) together with Ex. 12 impliesthat φA, ψA have a common eigen vector.
I By Ex. 11, this implies A has an eigen value inC.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 45: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/45.jpg)
I Ex. 13 Prove that S1(C, r) =⇒ S1(C, r + 1).
I Answer:Let n = 2km where m is odd and k ≤ r . LetA ∈ Mn(C). Then φA, ψA are two mutuallycommuting operators on Symn(C) which is ofdimension n(n + 1)/2 = 2k−1m(2km + 1) whichis not divisible by 2r .
I Therefore S1(C, r) together with Ex. 12 impliesthat φA, ψA have a common eigen vector.
I By Ex. 11, this implies A has an eigen value inC.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 46: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/46.jpg)
I Ex. 13 Prove that S1(C, r) =⇒ S1(C, r + 1).
I Answer:Let n = 2km where m is odd and k ≤ r . LetA ∈ Mn(C). Then φA, ψA are two mutuallycommuting operators on Symn(C) which is ofdimension n(n + 1)/2 = 2k−1m(2km + 1) whichis not divisible by 2r .
I Therefore S1(C, r) together with Ex. 12 impliesthat φA, ψA have a common eigen vector.
I By Ex. 11, this implies A has an eigen value inC.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 47: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/47.jpg)
I Ex. 13 Prove that S1(C, r) =⇒ S1(C, r + 1).
I Answer:Let n = 2km where m is odd and k ≤ r . LetA ∈ Mn(C). Then φA, ψA are two mutuallycommuting operators on Symn(C) which is ofdimension n(n + 1)/2 = 2k−1m(2km + 1) whichis not divisible by 2r .
I Therefore S1(C, r) together with Ex. 12 impliesthat φA, ψA have a common eigen vector.
I By Ex. 11, this implies A has an eigen value inC.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 48: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/48.jpg)
I Ex. 14 Conclude that every non constantpolynomial over complex numbers has a root.
I Answer:
I By Ex.3, it is enough to show that every n × ncomplex matrix has an eigen value.
I By Ex.9 this holds for odd n. This impliesS1(C, 1). By Ex. 13 applied repeatedly, we getS1(C, r) for all r . Write n = 2rm where m isodd. Then S1(C, r + 1) implies everyendomorphism A : Cn → Cn has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 49: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/49.jpg)
I Ex. 14 Conclude that every non constantpolynomial over complex numbers has a root.
I Answer:
I By Ex.3, it is enough to show that every n × ncomplex matrix has an eigen value.
I By Ex.9 this holds for odd n. This impliesS1(C, 1). By Ex. 13 applied repeatedly, we getS1(C, r) for all r . Write n = 2rm where m isodd. Then S1(C, r + 1) implies everyendomorphism A : Cn → Cn has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 50: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/50.jpg)
I Ex. 14 Conclude that every non constantpolynomial over complex numbers has a root.
I Answer:
I By Ex.3, it is enough to show that every n × ncomplex matrix has an eigen value.
I By Ex.9 this holds for odd n. This impliesS1(C, 1). By Ex. 13 applied repeatedly, we getS1(C, r) for all r . Write n = 2rm where m isodd. Then S1(C, r + 1) implies everyendomorphism A : Cn → Cn has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 51: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/51.jpg)
I Ex. 14 Conclude that every non constantpolynomial over complex numbers has a root.
I Answer:
I By Ex.3, it is enough to show that every n × ncomplex matrix has an eigen value.
I By Ex.9 this holds for odd n. This impliesS1(C, 1). By Ex. 13 applied repeatedly, we getS1(C, r) for all r . Write n = 2rm where m isodd. Then S1(C, r + 1) implies everyendomorphism A : Cn → Cn has an eigenvalue.
Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA
![Page 52: FUNDAMENTAL THEOREM of ALGEBRA through Linear Algebraars/TEP-fta.pdf · Linear Algebra at IITB, Aug. 2012 I We present a proof of Fundamental Theorem of Algebra through a sequence](https://reader030.vdocuments.mx/reader030/viewer/2022041101/5ed9515ff59b0f56f45f43aa/html5/thumbnails/52.jpg)
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Anant R. Shastri I. I. T. Bombay Derksen’s Proof of FUNDAMENTAL THEOREM of ALGEBRA