Notes on multilinear algebra

Citation for published version (APA):Blokhuis, A., & Seidel, J. J. (1983). Notes on multilinear algebra. (Eindhoven University of Technology : Dept ofMathematics : memorandum; Vol. 8312). Technische Hogeschool Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 1983-12

Augus t 1983

NOTES ON MULTILINEAR ALGEBRA

by

A. Blokhuis and J.J. Seidel

Eindhoven University of Technology

Deparcnent of Mathematics and Computing Science

PO Box 513, 5600 MB Eindhoven

The Netherlands

NOTES ON MULTILINEAR ALGEBRA

by

A. Blokhuis and J.J. Seidel

I. Symmetric functions

The elementary symmetric polynomials in d variables and their generating

function are defined by

Ak <.~) := L x. x. . .. x. for k :S d, zero for k >

jl< .. ·<jk J 1 J 2 J k

k d A t (~) := L Ak t n (1 + x. t)

k~O i=1 1

The homogeneous symmetric polynomials and their generating functions are

defined by

s (x) : = t -

L x. x .... x . • < < • J 1 J 2 J k J 1- " ·-Jk

Theorem 1.1. A /!) S -t (~) 1 •

Proof.

1 d d

A (x) n 1 - x. t n L

-t - i=l 1 i=l k~O

( 1 )

k (x. t) s (x) . 1 t-

d

- 2 -

Formula (1) generalizes in various ways. We have

where

A (V) s (V) t -t

( 2)

for the exterior powers Ak(V) and the symmetric powers Sk(V) of a vector

space V, whose definition will be recalled later.

A further generalization of (1) holds in the Grothendieck ring (R(G),e,~)

consisting of the G-modules V of a finite group G and, more abstractly,

in the theory of A-rings. Thus (2) follows from (1) by general abstract

nonsense.

References

s. Lang (1965), Algebra, page 431.

I.G. Macdonald (1979), Symmetric functions and Hall polynomials, Remark

2. 15, page 17.

D. Knutson (1973), A-rings and the representation theory of the symmetric

group, Springer Lect. Notes 308.

T. tom Dieck (1979), Transformation groups and representation theory,

Springer Lect. Notes 766.

R.P. Stanley, Combinatorial reciprocity theorems, M.e. tracts 56 (1974),

107 - 118 = Nij enrode Proceedings.

~Iore extended: Advanced Math. 14 (1974),194-253.

•

•

- 3 -

2. Tensor algebra

Let V denote a vector space of dim. d, with inner product (. ,.). The tensor

algebra TV, consisting of all tensors on V, is a graded algebra with

00

TV p (t) (\ _ d t) -\ .

The tensors of the form ~l ® ••• ® ~ form a basis for Tk V, and the inner

produc treads

(~l ® ••• ® ~ ':l.\ ® ••• ® ~) k

= n i=l

(x. ,v.) . -1 L.1

The exterior algebra AV, consisting of all skew symmetric tensors, is graded

with

AV

A basis for Ak is provided by the skew tensors

and the inner product reads

p( t) d

(\ + t) .

The symmetric algebra SV, consisting of all symmetric tensors, is graded with

00

SV p( t) -d

(\ - t) .

- 4 -

A basis for Sk ~s provided by the symmetric tensors

and the ~nner product reads

Let ~1""'~ denote any orthonormal basis for V. The corresponding ortho­

normal basis for Ak consists of the

k e- k I + ..• + kd k

The corresponding orthonormal basis for Sk consists of the

k .

k e-

/ k! k elements k' k' e- ,

I' ... d'

k kl, .•• ,kd Ell,

with e. ~ := e. v ... v e., k. times. Thus we use the same notation for both -~ -~ -~ ~

cases. It is convenient to abreviate k\! ... kd! by~! and k1 + '" + kd by

References

W.H. Greub, Multilinear Algebra, Springer \967.

R. Shaw, Linear Algebra and Group Representations I and II,

Academic Press 1982.

•

•

- 5 -

3. Exterior and symmetric powers of a matrix

Let A v ~ V denote a linear map of V. We define its k-th exterior power by

Ak (A) : Ak V ~ Ak V : ~I " ••• " ~ 1+ A ~I " ••• " A ~ ,

and its k-th symmetric power by

Sk V ~ Sk V ~ I v .•. v ~ 1+ A ~ I v ••. v A ~ •

We calculate the entries of the power matrices with respect to the standard

basis. We use the following notation, which applies for both Ak and Sk' For

~ and 3: wi th I~I = 13:1 = k the matrix A(~ 1.8:.) is the k x k matrix which is

obtained from the d x d matrix A by repeating k. times row i, and t. times ~ J

column j, for i,j 1,2, .•. ,d.

Theorem 3. I.

det A(~ 1 .8:.) per A(~ 1 3:)

Let A have the eigenvalues al,a 2 , ••• ,a

d. The eigenvalues of Ak(A) are the

(~) elementary, those of Sk(A) the (d +~ - I) homogeneous polynomials of

degree k in al, •.. ,ad .

Theorem 3.2.

det(1 + t A) Y. t k trace Ak (A) = I t I~I det A(~ 1 ~) k=O k

-I k _ 1 k 1 l'er A(~ 1 ~) det (I - tA) I t trace Sk(A) - kL t- k!

k~O

- 6 -

Proof. The first result simply amounts to

deter + t A) I + t

d

I i=1

2 a .. + t

11 I i<j

a .. 11 a ..

1J d + ..• + t det A .

a .. a .. J1 JJ

Both results are easily proved by considering the eigenvalues on both sides,

and by using the proof of Theorem 1.1.

For the generating functions

the theorem implies

-I deter + t A) = trace !I. (A) = trace S (A)

t -t

and I A t (~) s _ t (~), 1n agreement wi th Theorem I. I .

References

C.C. MacDuffee, The theory of matrices, Chelsea (1946).

M. Marcus, H. Mine, A survey of matrix theory (1964).

H. Mine, Permanents, Addison-Wesley (1978).

H. Mine, Theory of permanents 1978-1981, Lin. Multilin. Alg. 12 (1983)

227 - 263.

•

•

- 7 -

4. MacMahon's master theorem

Q, k Lemma 4.1. The coefficient of x- 1n (A x)- equals

Q,I! per A(~ I .~) .

Proof. Convince yourself by writing out A(~ I!) in block form.

k k _Th_e_o_r..;.e_m_4..;. • ..;...2 (MacMahon). The coefficient of x- in (A ~)- equals the coefficient

of xk in

l/det(1 - Af.,(~)), where f.,(x)

Proof (I.G. Macdonald). Lemma 4.1 and Theorem 3.1 imply that the coefficient

of ~~ in (A ~)~ equals the (~,~)-entry in the k-th symmetric power Sk (A),

where k = I~I. Hence it is the coefficient of ~ in trace Sk(Af.,(~)). But

I trace Sk (A f., (~)) k~O

1/ de t (I - A f., (~))

5. Bebiano's formula

Theorem 5.1.

exp(~,A X) t

Proof. The formula

k (~,A y)

k!

oo

L tk

k=O

- 8 -

I~I =f!1 =k

k x- R­y-

k R-x- r-k! "iT per

k! ~! per A(~ I .8:)

A(~ I!)

~s obtained by taking the inner products of the symmetric tensors on the

left and on the right hand sides of the following formulae

I!t=k

Indeed, we have

(x v ... v ~, ~ v .•. v z) (x ~ ® z)

Reference

N. Bebiano, Pac. J. Math. 101 (1982), 1-9.

•

o

- 9 -

6. Fredholm's determinant

Fredholm's integral equation

1

u(x) f(x) + A f K(x,t) u(t) dt

o

1S approximated by the set of matrix equations

(I - AM )u = f d- -

d 1,2, ..• ,

as follows. The interval [O,IJ is devided into d equal parts by

1 d - 1 o < d < '" < -d- < I, and! (= !d) i

has components fi = fed)' whereas Md

-1 i 1.' has entries Md(i,j) = d K(-d' d)'

Fredholm's determinant 1S defined as follows:

I

I - A J o

1

K(t,t)dt + ~~ f f o 0

K(tl,t l )

K(t 2 ,t))

+;~ fff+···· It is the limi t, for d + 00, of

K(t l ,t2)

K(t2,t

2)

d d + (- 1 ) A x de t M d

- 1{) -

As a consequence of (3) and (4) we have

This is in agreement with (2). Thus, Fredholm's equation may be solved in

terms of permanents.

References

W.V. Lovitt, Linear integral equations, Dover 195.

N.G. de Brui jn, The Lagrange-Good inversion formula and its application to

integral equations, J. Math. Anal. Appl. ~ (1983),397-409.

D. Kershaw, Permanents in Fredholm's theory of integral equations, J. Integral

Equ. (1979) 281 - 290.