Invariant Determinants and Differential Forms

Transcription

1 Invariant Determinants and Differential Forms IITG PHYSICS SEMINAR Srikanth K.V. Department of Mathematics, IIT Guwahati Invariant Determinants and Differential Forms p. 1

2 Overview of the talk 1. Laplace s formula and Leibnitz s formula 2. Properties of determinants 3. Determinant of a linear operator 4. A dilemmatic example 5. Volume in Euclidean spaces 6. Abstract volume functions in vector spaces 7. Geometric definition of determinant function 8. Advantages of geometric definition 9. Differential forms Invariant Determinants and Differential Forms p. 2

3 Laplace s and Leibnitz s formula Let A be an n n matrix over a field F. Then deta = = n a ij C ij i=1 n ( 1) i+j a ij M ij i=1 where C ij is cofactor of a ij and M ij is the (ij)-th minor of A. Invariant Determinants and Differential Forms p. 3

4 Laplace s and Leibnitz s formula Let A be an n n matrix over a field F. Then deta = σ S n sign(σ) n i=1 a iσ(i) This formula probably has its roots in Cramer s Rule type solutions to a linear system Ax = b. Remark: This formula makes sense even if F is a commutative ring. Invariant Determinants and Differential Forms p. 4

5 Properties of determinants 1. det(ab) = det A det B 2. If any row/column of A is a linear combination of the remaining rows/columns of A then det A = If B is a matrix obtained by interchanging any two rows or columns of matrix A then det A = det B 4. det(ka) = k n det(a), for any k F 5. det A = deta T 6. deta 1 = (deta) 1 Invariant Determinants and Differential Forms p. 5

6 Properties of determinants Remarks: The proofs of properties 1, 2 and 3 are usually via the defining formulae. They are somewhat hard and unilluminating. The invariant or geometric definition we are going to see makes the proofs of these a triviality. Invariant Determinants and Differential Forms p. 6

7 Determinant of a linear operator We know that determinants of similar matrices are equal. Invariant Determinants and Differential Forms p. 7

8 Determinant of a linear operator We know that determinants of similar matrices are equal. If f is a linear operator on a vector space V the following is well defined det f = det (matrix of f with respect to any basis B of V) Invariant Determinants and Differential Forms p. 7

9 Determinant of a linear operator We know that determinants of similar matrices are equal. If f is a linear operator on a vector space V the following is well defined det f = det (matrix of f with respect to any basis B of V) The above definition does not tell us which property of f is captured by detf Invariant Determinants and Differential Forms p. 7

10 A dilemmatic example Invariant Determinants and Differential Forms p. 8

11 Volume in euclidean spaces Invariant Determinants and Differential Forms p. 9

12 Abst. vol. functions in vector spaces Let V be a vector space over a field F. Define a k-volume function as a function µ : V k F which satisfies 1. µ is k-linear 2. µ is alternating i.e. µ(a 1,a 2,...,a k ) = 0,whenevera i = a j fori j, 1 i,j k Invariant Determinants and Differential Forms p. 10

13 Abst. vol. functions in vector spaces Let V be a vector space over a field F. Define a k-volume function as a function µ : V k F which satisfies 1. µ is k-linear 2. µ is alternating i.e. µ(a 1,a 2,...,a k ) = 0,whenevera i = a j fori j, 1 i,j k µ is alternating µ is skew symmetric. If {a i } k i=1 is a linearly dependent set then µ(a 1,...,a k ) = 0 Invariant Determinants and Differential Forms p. 10

14 Abst. vol. functions in vector spaces Note that if σ S k, the symmetric group over k symbols, then µ(a σ(1),a σ(2),...,a σ(k) ) = sign(σ)µ(a 1,a 2,...,a k ) Invariant Determinants and Differential Forms p. 11

15 Abst. vol. functions in vector spaces Note that if σ S k, the symmetric group over k symbols, then µ(a σ(1),a σ(2),...,a σ(k) ) = sign(σ)µ(a 1,a 2,...,a k ) Let dimv = n. For any k N, define Λ k (V) = {µ : V k F : µisk linearandalternating} Invariant Determinants and Differential Forms p. 11

16 Abst. vol. functions in vector spaces Note that if σ S k, the symmetric group over k symbols, then µ(a σ(1),a σ(2),...,a σ(k) ) = sign(σ)µ(a 1,a 2,...,a k ) Let dimv = n. For any k N, define Λ k (V) = {µ : V k F : µisk linearandalternating} It can be verified that Λ k (V) is a vector space over F and dimλ k (V) = ( n k ) ifk n 0 if k > n Invariant Determinants and Differential Forms p. 11

17 Determinant using volume functions Since Λ n (V) is one dimensional, any two (non zero) volume functions on V are multiples of each other. Invariant Determinants and Differential Forms p. 12

18 Determinant using volume functions Since Λ n (V) is one dimensional, any two (non zero) volume functions on V are multiples of each other. Let f : V V be a linear transformation and {b i } n i=1 a basis of V. Pick any 0 µ Λn (V). Invariant Determinants and Differential Forms p. 12

19 Determinant using volume functions Since Λ n (V) is one dimensional, any two (non zero) volume functions on V are multiples of each other. Let f : V V be a linear transformation and {b i } n i=1 a basis of V. Pick any 0 µ Λn (V). Observation: The number µ(f(b 1 ),...,f(b n )) µ(b 1,...,b n ) is independent of both the choice of basis and the choice of µ. Invariant Determinants and Differential Forms p. 12

20 Geometric defn. of the det. function So define det f as det f = µ(f(b 1),...,f(b n )) µ(b 1,...,b n ) using any basis {b i } n i=1 µ Λ n (V) of V and any nonzero Invariant Determinants and Differential Forms p. 13

21 Geometric defn. of the det. function So define det f as det f = µ(f(b 1),...,f(b n )) µ(b 1,...,b n ) using any basis {b i } n i=1 µ Λ n (V) of V and any nonzero Remark: The above definition is faithful to the interpretation of determinant of a linear transformation as "volume expansion factor" Note that if h is a singular transformation, then det h = 0. Invariant Determinants and Differential Forms p. 13

22 Consistency with known defn. Given n vectors b 1,...,b n in V, write them as a column B. Express the entries in terms of the standard basis E via B = CE. The usual determinant of the matrix C can be viewed as the value of µ(b 1,...,b n ) for an element µ in Λ n (V). Using this one can prove consistency. Invariant Determinants and Differential Forms p. 14

23 Determinant of a composition If f or g is singular then g f is singular and det f = 0 or detg = 0 and det g f = 0. So det(g f) = det g det f. Invariant Determinants and Differential Forms p. 15

24 Determinant of a composition If f or g is singular then g f is singular and det f = 0 or detg = 0 and det g f = 0. So det(g f) = det g det f. If f and g are nonsingular then det(g f) = µ(g f(b 1),...,g f(b n ) µ(b 1,...,b n ) = µ(g(f(b 1)),...,g(f(b n )) µ(f(b 1 ),...,f(b n ) µ(f(b 1 ),...,f(b n )) µ(b 1,...,b n ) = det g det f Invariant Determinants and Differential Forms p. 15

25 Search for Integrands Question: If N is a k-dimensional submanifold of a differential manifold M, what is the right object which we can integrate over N? Invariant Determinants and Differential Forms p. 16

26 Search for Integrands Question: If N is a k-dimensional submanifold of a differential manifold M, what is the right object which we can integrate over N? Hint: We have encountered a similar situation in vector calculus when we studied line integrals (like work) and surface integrals (like flux). Invariant Determinants and Differential Forms p. 16

27 Search for Integrands Question: If N is a k-dimensional submanifold of a differential manifold M, what is the right object which we can integrate over N? Hint: We have encountered a similar situation in vector calculus when we studied line integrals (like work) and surface integrals (like flux). Problem: Such integrals assume inner product on tangent spaces Invariant Determinants and Differential Forms p. 16

28 Differential forms Observation: In line integrals, integrand :: linear function & In surface integrals, integrand :: bilinear alternating function (on tangent spaces) Invariant Determinants and Differential Forms p. 17

29 Differential forms Observation: In line integrals, integrand :: linear function & In surface integrals, integrand :: bilinear alternating function (on tangent spaces) Pick a smoothly varying ω on M such that for each p M, ω p Λ k (T p M) where k is the dimension of the submanifold N. Invariant Determinants and Differential Forms p. 17

30 Differential forms Observation: In line integrals, integrand :: linear function & In surface integrals, integrand :: bilinear alternating function (on tangent spaces) Pick a smoothly varying ω on M such that for each p M, ω p Λ k (T p M) where k is the dimension of the submanifold N. Such a function is called a differential k form. Invariant Determinants and Differential Forms p. 17

31 Generalizations Grad, Curl and Divergence :: Exterior derivative Theorems of Green, Gauss and Stokes :: General Stokes Theorem Path independence and potential functions :: Poincare Lemma Failure of Poincare Lemma :: Cohomology groups Relation between Cohomology groups and Shape of M :: Theorem of de Rham Invariant Determinants and Differential Forms p. 18