We give a short proof of the Cauchy-Binet determinantal formula using
multilinear algebra by first generalizing it to an identity {\em not} involving
determinants. By extending the formula to abstract Hilbert spaces we obtain, as
a corollary, a generalization of the classical Parseval identity.Comment: 9 pages, 2 diagram

Konstantopoulos, Takis

English

arXiv

We give a short proof of the Cauchy-Binet determinantal formula using
multilinear algebra by first generalizing it to an identity {\em not} involving
determinants. By extending the formula to abstract Hilbert spaces we obtain, as
a corollary, a generalization of the classical Parseval identity

University of Liverpool Repository

arXiv:1305.0644v1  [math.RA]  3 May 2013A multilinear algebra proof of the Cauchy-Binetformula and a multilinear version of Parseval’sidentityTakis Konstantopoulos1 May 2013AbstractWe give a short proof of the Cauchy-Binet determinantal formula using multilinearalgebra by first generalizing it to an identity not involving determinants. By extendingthe formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of theclassical Parseval identity.Keywords and phrases. Cauchy-Binet theorem, determinant, matrix identities, Hilbertspace, Parseval’s identity, multilinear algebra, exterior products, projections,Pythagorean theorem, Fock spaceAMS 2000 subject classifications. Primary 15A15,15A69; secondary 15A24,46C051 Introduction and overviewThe classical Cauchy-Binet formula states that if A,B are two matrices over R (or any field)of sizes n×N , N × n, respectively, with n ≤ N , thendet(AB) =∑σdet(Aσ) det(Bσ) (1)where the sum is taken over all σ = (σ1 < σ2 < · · · < σn), with σi ∈ {1, . . . , N}, and whereAσ (respectively Bσ) is the n×n submatrix of A (respectively submatrix of B) obtained bydeleting all columns (respectively all rows) except those with indices in σ.There are many proofs of this formula, each telling its own story, explaining the formulafrom a different point of view. The most direct way of proving the formula is by writing downthe determinant as a sum over permutations and performing algebraic manipulations. Thisis the approach taken in many linear algebra books; see, e.g., Marcus and Minc [8, Theorem6.1, p. 128] and Gohberg et al. [7, Theorem A.2.1, p. 651]. A probabilistic interpretation andproof of the formula (which starts by using the formula for a determinant) is also available[4, 5]. On the other hand, there are many combinatorial proofs. Suffice, perhaps, to refer to1the one chosen to be included in the “Proofs from The Book” [2] by Aigner and Ziegler. Thisis a nice proof (after all, it is a proof from The Book) based on the beautiful Gessel-Vienotlemma which states that, in a finite weighted acyclic directed graph, the determinant of thepath matrix between two sets of vertices of cardinality n each equals a sum over all possiblevertex-disjoint path systems; see [2, Chap. 29, p. 196] and [1] for details. Another verysimple proof appears in the recent book by Terence Tao [13, p. 298]) on random matrices.This proof is based on a relation between the characteristic polynomials of AB and BA.On the other hand, it is well-known that the Cauchy-Binet formula is a generalization ofthe Pythagorean theorem. Indeed, let A be a n×N real matrix, n ≤ N , and take B = AT ,the transpose of A. Since Bσ = (AT )σ = (Aσ)T , the formula givesdet(AAT ) =∑σdet(Aσ)2,which can be interpreted geometrically as follows: The parallelotope in RN generated bythe n row vectors of A has n-dimensional Lebesgue measure√det(AAT ). Therefore theformula says that the square of the n-dimensional measure of an n-dimensional parallelotope,embedded in a higher-dimensional Euclidean space, equals the sum of the squares of themeasures of its projections onto all possible n-dimensional coordinate hyperplanes. If n = 1this reduces to the Pythagorean theorem.The goal of this short article is to give a proof of the Cauchy-Binet formula which isas simple as possible, from an algebraic-geometric viewpoint. If n = 1, the Cauchy-Binetformula is a triviality: it states that the inner product of two N -dimensional vectors equalsthe sum of the products of their components:(a1 . . . , aN ) · (b1, . . . , bN )T =N∑σ=1aσbσ.There is no need to take determinants here, because both sides involve 1× 1 matrices, i.e.,real numbers. What we show is that the general case, when n ≥ 1, is the same, but onbigger vector spaces. In Section 2 we give an account of the ingredients we need, and, inSection 3, we state and prove the main formula (Theorem 1) without determinants and in amore general setup; a corollary of it is the classical Cauchy-Binet formula. Then, in Section4, we see that the formula can be extended to a Hilbert space, giving a generalization of theclassical Parseval identity. We conclude with a few bibliographic remarks.2 The main ingredientsThe main theorem, Theorem 1 below, is requires two ingredients.(i) The first is the notion of the determinant of a linear transformation F : X //X ona vector space X of dimension d. The dimension of the linear space∧mX of alternatingm-linear maps ω : Xm // R is(dm). For each m, the m-th level dual F ∗ :∧mX // ∧mX2of F is defined byF ∗ω[x1, . . . , xm] := ω[Fx1, . . . , Fxm]. (2)See, e.g., [12]. (Duals obey the standard composition rules: (GF )∗ = F ∗G∗.) Since∧dX is1-dimensional, the d-th level dual F ∗ is multiplication by a constant. This constant is, bydefinition, the determinant of F :F ∗ω = (detF ) · ω, ω ∈∧dX. (3)(ii) The second ingredient is very simple too. Let X,Y,Z be vector spaces, and F :X //Y , G : Y //Z linear maps. Suppose Y is the direct sum of Y1, . . . , YK . Let Pi : Y //Yi,1 ≤ i ≤ K, be the projections corresponding to this direct sum (so idV = P1 + · · · + PK isa partition of the identity on V ), and let Ei : Yi // Y be the natural embedding of Yi intoY . Then, clearly,GF =K∑i=1(GEi)(PiF ). (4)See Diagram 1.3 An abstract version of the Cauchy-Binet formulaLet U, V,W be finite-dimensional vector spaces of arbitrary dimensions, and let B : U //V ,A : V // W be two linear maps. Fix n ∈ N and consider the n-th level duals B∗ :∧n V // ∧n U , A∗ : ∧nW // ∧n V . Let N be the dimension of V and let f1, . . . , fN bea basis for V . See Diagram 2. Denote by Sn(N) the set of subsets of {1, . . . , N} of size n.For each σ ∈ Sn(N), let Vσ be the subspace of V spanned by {fi, i ∈ σ} and consider thedirect sumV = Vσ ⊕ Vσ, (5)where σ := {1, . . . , N} \ σ, lettingPσ : V // Vσbe the projection of V onto Vσ along Vσ, andEσ : Vσ // Vthe natural embedding of Vσ into V .Theorem 1.(AB)∗ =∑σ∈Sn(N)(PσB)∗(AEσ)∗, (6)Proof. The(Nn)–dimensional space∧n V is the direct sum of the 1-dimensional spaces∧n Vσ,where σ ranges in Sn(N): ∧n V =⊕σ∈Sn(N)∧n Vσ. (7)3Let Pσ :∧n V // ∧n Vσ be projections corresponding to this direct sum, and let Eσ :∧n Vσ //∧n V be natural embedding. Using (4) (with A∗, B∗ in place of F , G, respectively,and K =(Nn)), we haveB∗A∗ =∑σ∈Sn(N)(B∗Eσ)(PσA∗).See Diagram 2. Since (see Lemma 1 below)Eσ = P∗σ , Pσ = E∗σ,the theorem follows from the composition rules of the duals.Lemma 1.Pσ = E∗σ, Eσ = P∗σ .Proof. We identify Sn(N) with the set of strictly increasing sequences of length n withvalues in {1, . . . , N}. Thus, if σ is a subset of {1, . . . , N} we let (σ1, . . . , σn) be a listing ofits elements in increasing order. To prove the first equality it suffices to show thatPσω[v1, . . . , vn] = ω[Eσv1, . . . , Eσvn],for all ω ∈ Λn(V ) and all v1, . . . , vn ∈ Vσ. But then Eσvi = vi and, since Vσ is spanned byfσ1 , . . . , fσn , it suffices to show thatPσω[fσpi(1) , . . . , fσpi(n) ] = ω[fσpi(1) , . . . , fσpi(n) ],where pi is a permutation of {1, . . . , n}. Since ω =∑τ∈Sn(N)Pτω [this is the partition ofthe identity on∧n V corresponding to (7)] we may replace ω by Pτω in the last display:PσPτω[fσpi(1) , . . . , fσpi(n) ] = Pτω[fσpi(1) , . . . , fσpi(n) ].But then, if τ = σ the two sides are obviously equal, and if τ 6= σ the left-hand side equalszero and Pτω[fσpi(1) , . . . , fσpi(n) ] = 0.To prove the second equality it suffices to show thatEσω[v1, . . . , vn] = ω[Pσv1, . . . , Pσvn],for all ω ∈ Λn(Vσ) and all v1, . . . , vn ∈ V . But then Eσω = ω. Since vi = Pσvi + Pσvi[corresponding to (5)], we haveEσω[v1, . . . , vn] = ω[Pσv1 + Pσv1, . . . , Pσvn + Pσvn].Using the multilinearity of ω we split the latter into 2n terms, all of which are zero exceptthe one involving only Pσvi as arguments.Consider now the case where W = U . Moreover, take the number n in Theorem 1 tobe equal to their common dimension. Assume n ≤ N = dimV to avoid trivialities. Thenthe linear maps (AB)∗, (PσB)∗, and (AEσ)∗, appearing in formula (6), are maps between41-dimensional spaces. Since the spaces Vσ and U have common dimension n, we can identifythem by means of a linear bijectionϕσ : Vσ // U.Then(PσB)∗(AEσ)∗ = (PσB)∗ϕ∗σ(ϕ−1σ )∗(AEσ)∗ = (ϕσPσB)∗(AEσϕ−1σ )∗,and so(AB)∗ =∑σ∈Sn(N)(ϕσPσB)∗(AEσϕ−1σ )∗. (8)Since all three linear maps AB, ϕσPσB, AEσϕ−1σ are linear maps on the same 1-dimensionalvector space U , it follows, from the definition of the determinant, thatdet(AB) =∑σ∈Sn(N)det(ϕσPσB) det(AEσϕ−1σ ). (9)(The role of ϕσ is to force all maps be on the same space, so we can talk about determinants.)In the case where U = Rn, V = RN , this proves the classical Cauchy-Binet formula (1).If N = n, then we have shown that the determinant of the product is the product of thedeterminants.Therefore (1) follows from (9). The latter is a restatement of (8). But (8) is a specialcase of (6) because in (6) we allow U, V,W to be different with dimensions that may bedistinct from n.4 Multilinear Parseval’s identityWe are now going to replace the middle space V of the previous setup by a separableHilbert space H over the complex numbers C, having inner product 〈x, y〉. Let f1, f2, . . .be an orthonormal basis for H. Let∧nH be the collection of all continuous alternatingmultilinear functionals ω : Hn // C. In particular,∧1H = H∗ is the Hilbert space dualof H. By the Riesz-Fischer theorem, f1, f2, . . . forms a basis for∧1H in the sense thatevery ω ∈∧1H can be uniquely written as ω[x] = ∑∞σ=1 aσ〈fσ, x〉, for aσ ∈ C such that∑σ |aσ|2 < ∞. More generally,∧nH is a separable Hilbert space with orthonormal (withrespect to a suitably defined inner product) basisfσ1 ∧ · · · ∧ fσn , σ = (σ1, . . . , σn) ∈ Sn(N),where Sn(N) is the collection of all n-tuples (σ1, . . . , σn) of positive integers such that σ1 <· · · < σn. Recall that the wedge product satisfies, by definition,(f1 ∧ f2)[x, y] = f1[x]f2[y]− f1[y]f2[x],and, more generally, fσ1∧· · ·∧fσn is obtained by antisymmetrization of the tensor product offσ1 , . . . , fσn . Incidentally, the direct sum of⊕∞n=0∧nH (where ∧0H := C) is the so-called5alternating Fock (or fermionic) space [11]. Wedge products can be defined, by linearity,between any finite number of elements of this space.If H1,H2 are two Hilbert spaces and F : H1 //H2 is a continuous linear function thenF ∗ :∧nH2 // ∧nH1 is defined as before–see (2)–and is, moreover, continuous.Theorem 2. Let H be a separable Hilbert space over C with orthonormal basis f1, f2, . . .,and let n be a positive integer. For each σ ∈ Sn(N), let Hσ be the subspace spanned byfσ1 , . . . , fσn . Let Eσ : Hσ //H be the natural embedding of Hσ into H and Pσ : H //Hσthe orthogonal projection of H onto Hσ. If U , W are finite-dimensional vector spaces overC and B : U //H, A : H //W continuous linear maps, then(AB)∗ =∑σ∈Sn(N)(PσB)∗(AEσ)∗.If W = U with common dimension n, and if ϕσ : Hσ // U is any linear bijection, then(AB)∗ =∑σ∈Sn(N)(ϕσPσB)∗(AEσϕ−1σ )∗.In particular,det(AB) =∑σ∈Sn(N)det(ϕσPσB) det(AEσϕ−1σ ).The proof of this theorem is exactly as in the finite-dimensional case. Infinite sums haveto be understood in the Hilbert space sense.Consider now H = L2[0, 1] with inner product 〈x, y〉 =∫ 10 x(t)y(t)dt and the standardorthonormal basis ek(t) = exp(i2pikt), k ∈ Z, and let U = W = Cn, for a given positiveinteger n. A continuous linear map A : L2[0, 1] // Cn is necessarily (Riesz representationtheorem) of the formAx =(〈x, a1〉, . . . , 〈x, an〉)=(∫ 10a1(t)x(t)dt, . . . ,∫ 10an(t)x(t)dt), x ∈ L2[0, 1],where a1, . . . , an ∈ L2[0, 1]. A linear map B : Cn // L2[0, 1] is of the form(Bu)(t) = u1b1(t) + · · ·+ unbn(t), u ∈ Cn,where b1, . . . , bn ∈ L2[0, 1]. Hence the jk-entry of the matrix of AB : Cn //Cn, with respectto the standard basis on Cn, is given by(AB)jk =∫ 10aj(t)bk(t)dt.Consider now σ ∈ Sn(Z), i.e., σ = (σ1, . . . , σn) ∈ Zn with σ1 < · · · < σn. (There is nodifficulty in replacing N in the above theorem by Z.) Then Hσ is the subspace of L2[0, 1]spanned by eσ1 , . . . , eσn . So the orthogonal projection Pσ : H//Hσ is given byPσx = x̂(σ1)eσ1 + · · ·+ x̂(σn)eσn ,6wherex̂(k) :=∫ 10x(t) exp(−i2pikt)dt, k ∈ Z,are the Fourier coefficients of x. Letting ϕσ : Hσ // Cn be the linear bijection that takeseσr into the r-th standard basis vector of Cn, for r = 1, . . . , n, we see that the jk-entry ofthe matrix of ϕσPσB is(ϕσPσB)jk = b̂k(σj).Arguing analogously, the jk-entry of the matrix of AEσϕ−1σ is(AEσϕ−1σ )jk = âj(σk).Hence the last formula of Theorem 2 givesdet1≤j,k≤n∫ 10aj(t)bk(t)dt =∑σ∈Sn(Z)det1≤j,k≤n[âj(σk)]det1≤j,k≤n[̂bj(σk)]=1n!∑σ1∈Z· · ·∑σn∈Zdet1≤j,k≤n[âj(σk)]det1≤j,k≤n[b̂j(σk)],where the second equality follows from the fact that applying the permutation of (σ1, . . . , σn)to both matrices will change the sign of both determinants simultaneously and the fact thatrepeated indices result into zero determinants. For n = 1, this is the standard Parsevalidentity.Of course, there is nothing special with the Lebesgue measure. We can obtain formulasfor any other L2 space or other separable Hilbert spaces.5 RemarksMy motivation for this article was due to my desire to understand some elements of randommatrix theory [3] and determinantal point processes [6]. In particular, the derivation of theubiquitous Tracy-Widom probability distribution [3] involves several applications of Cauchy-Binet type formulas. When I looked at it first, a standard computational proof was not toosatisfactory. I discovered that there are many proofs, which can be roughly classified intocombinatorial and algebraic ones. The version presented in this short article was inspired bythe simple observation that the Cauchy-Binet formula is a version of Pythagorean theorem:it is a version of the Pythagorean theorem on∧nRN , with n ≤ N (which is of courseisomorphic to R(Nn)).Several years ago, Zeilberger [14] “complained” that, to most contemporary mathemati-cians, matrices and linear transformations are practically interchangeable notions and thatthe mainstream ‘Bourbakian’ establishment, with its profound disdain for the concrete, goesas far as to frown at the mere mention of the word ‘matrix’. He then explains how “to [him],as well as to other ‘dissidents’ called ‘combinatorialists’, a matrix has nothing whatsoeverto do with that intimidating abstract concept called ‘a linear transformation between linear7vector spaces’ ” and, by thinking of matrices as putting weights on a graph, he developsa combinatorial way of interpreting and proving fundamental results such as the Cayley-Hamilton theorem. The Cauchy-Binet formula has found a nice proof, in the Zeilbergersense, as a corollary of the Gessel-Vienot lemma. We also mention Zeng’s proof [15] whichalso uses Zeilberger’s methods.In a sense then, what we have done here is in exactly the opposite of Zeilberger’s spirit,because the proof presented uses nothing else but the concept of a linear map between vectorspaces (and lots of definitions). Each point of view has its own merits in that, for instance,it leads to different kind of extensions. (Extensions to infinite matrices are not easy whenthe combinatorial point of view is adopted.)We finally remark that there are generalizations of the Cauchy-Binet formula for the casewhere the matrices contain elements of a noncommutative ring [9]. We do not know how toextend the ideas above to this case.AcknowledgmentsI thank Svante Janson [10] for pointing out reference [9] to me and for his comments on thisarticle, and Richard Ehrenborg [4] for kindly making his notes available to me.References[1] Martin Aigner. Lattice paths and determinants. Comput. discrete mathematics, 1-12, Lect.Notes Comp. Sci. 2122, Springer, Berlin, 2001.[2] Martin Aigner and Gu¨nter Ziegler. Proofs from the Book. Springer, 4th ed., New York, 2009.[3] Greg Anderson, Alice Guionnet and Ofer Zeitouni. An Introduction to Random Matrices.Cambridge Univ. press, 2009.[4] Richard Ehrenborg. Personal communication, 2013.[5] Richard Ehrenborg and Gian-Carlo Rota. Topics on polynomials. Lecture Notes, MIT, 1991.[6] John BenHough, ManjunathKrishnapur, Yuval Peres and BalintVirag. Zeros of GaussianAnalytic Functions and Determinantal Point Processes. American Math. Society, 2009.[7] Israel Gohberg, Peter Lancaster and Leiba Rodman. Invariant Subspaces of Matrices withApplications. SIAM Classics in Applied Mathematics 51, Wiley, New York, 2986.[8] Marvin Marcus, Henryk Minc. Introduction to Linear Algebra. MacMillan, New York, 1965;Reprinted by Dover, New York, 1965.[9] Sergio Caracciolo, Alan Sokal and Andrea Sportiello. Noncommutative determinants,Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turn-bull identities. Electron. J. Combin. 16, no. 1, Research Paper 103, 43 pp., 2009.[10] Svante Janson. Personal communication, 2012.[11] Michael Reed and Barry Simon. Methods of Modern Mathematical Physics, Vol. II. AcademicPress, 1975.8[12] Michael Spivak. Calculus on Manifolds. Addison-Wesley, 1965.[13] Terence Tao. Topics in Random Matrix Theory:http://terrytao.wordpress.com/books/topics-in-random-matrix-theory/[14] Doron Zeilberger. A combinatorial approach to matrix algebra. Discr. Math. 56, 61-72, 1985.[15] Jiang Zeng. A bijective proof of Muir’s identity and the Cauchy-Binet formula. Linear Alg.Appl. 184, 79-82, 1993.Takis KonstantopoulosDepartment of MathematicsUppsala University751 06 UppsalaSwedentakis@math.uu.sewww.math.uu.se/∼takisX YF // ZG //PiOOEiYiY =K⊕i=1YiGF =K∑i=1(GEi)(PiF )Diagram 1W VooAUooBOOEσ PσVσ∧nW∧nVA∗// ∧n UB∗ //PσOOEσ∧nVσ∧nV =⊕σ∈Sn(N)∧nVσB∗A∗ =∑σ∈Sn(N)(B∗Eσ)(PσA∗)Diagram 29

A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity

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