Notes on multilinear algebra

Notes on multilinear algebra

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Blokhuis, A., & Seidel, J. J. (1983). Notes on multilinear algebra. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8312). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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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

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 J2 Jk

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- " ·-J_k 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

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) (\ + t) d .

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

00

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, .•• ,k d 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 a

l,a2, ••• ,ad. The eigenvalues of Ak(A) are the

(~)

elementary, those of Sk(A) the (d

+~

-

I)

homogeneous polynomials of degree k in al, •.. ,a

d.

Theorem 3.2.

det(1 + t A) Y. tk 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!

Proof. The first result simply amounts to deter + t A) I + t d

I

i=1 2 a .. + t 11 _i<j

I

a .. 11 a .. 1J _{+ ..• + t} d _{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

5. Bebiano's formula

Theorem 5.1.

exp(~,A X) t

Proof. The formula

k (~,A y) k! oo

L

tk k=O

I~I

_{=f!1 =k} k x- y- R-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.

•

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) has components fi = fed)' whereas Md i

-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(t₂,t))

+;~

fff+····

It is the limi t, for d +

00,

of

K(t l,t2) K(t 2,t2) d d + (-1 ) A x de t M d

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 Bruijn, 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