Department

MathematicsCourse Code

MAT224H1Professor

Martin, BurdaThis

**preview**shows pages 1-2. to view the full**8 pages of the document.**Lecture 13 notes byY. Burda

1Reminder about dotproduct onRn

Recall thatonRnthe dot product of vectorsv=x1

.

.

.

xnand w=y1

.

.

.

ynis

deﬁned bythe equation

v·w=x1y1+. . . +xnyn

It satisﬁes the following properties:

Linearity:(c1v1+c2v2)·w=c1v1·w+c2v2·w

Symmetry:v·w=w·v

Positivity:v·v>0for v6=0

2Innerproduct spaces

The dotproduct onRnis goodfor doing geometry in n-dimensional space:

measuring lengths, angles, areasand so on. But this is more or less all

it is goodfor.However therearesimilar notions on othervectorspaces,

e.g. spaces of functions in analysis or random variables in statistics. These

notions allowus to answer questions “howclosearetwofunctions toeach

other”, “howcorrelatedare twoevents” and so on. Wewill nowdeﬁne what

an inner product is as ageneralization of the notion of the dot-product on

Rn:

Deﬁnition. Aninner product on areal vector spaceVis away to assign a

number (v,w)to any pair of two vectorsv,w∈Vso that the following three

conditions hold:

Linearity:(c1v1+c2v2,w)=c1(v1,w)+c2(v2,w)

Symmetry:(v,w)=(w,v)

Positivity:(v,v)>0for v6=0

Notethat symmetry and linearityin the ﬁrst argumentimply linearityin

the second parameter as well: (v,c1w1+c2w2)=c1(v,w1)+c2(v,w2)

Example:on the space P1(R)of degree one polynomial the formula

(p(x),q(x)) =p(0)q(0)+p(3)q(3)deﬁnes an inner product.

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Indeed, linearityand symmetry are easytocheck.Forpositivitynote

that (p(x),p(x)) =p(0)2+p(3)2≥0. Wehaveto showthatthe equalityis

possible in this inequalityonly if pis the zero polynomial. Whatwesee is

thatif (p(x),p(x)) =0then p(0)=p(3)=0. For a general polynomial this

is not enoughto deduce that pis the zero polynomial. However, since we

knowthat pis linear and has tworoots, we candeduce that it is identically

zero.

3Innerproduct in coordinates

Suppose that (−,−)is ainner product on Vand A=(u1,...,un)is abasis

of V.

Wewantto ﬁnd the inner productofvand wgiven their coordinates

reltiveto A:v=x1u1+. . . +xnunand w=y1u1+. . . +ynun.

Todo sowewill use linearityofinner products:

(v,w)=(x1u1+. . . +xnun,w)=x1(u1,w)+. . . +xn(un,w)=

=x1(u1,y1u1+. . . +ynun)+. . . +xn(un,y1u1+. . . +ynun)=

=X

i,j=1..n

xiyj(ui,uj)

Wecan write the result as (v,w)=(x1... xn) (u1,u1)... (u1,un)

.

.

.....

.

.

(un,u1)... (un,un)!y1

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.

.

ynor

(v,w)=XTAY

where X=[v]A,Y=[w]Aand Ais the matrix whose (i, j)-th entry is (ui,uj).

Wecall this matrix Athe matrix of the inner product (−,−)relativeto

basis A.

Example:The matrix of the inner product (p(x),q(x)) =p(0)q(0)+

p(3)q(3)relativeto the basis (1,x)of P1(R)is (1,1) (1,x)

(x,1) (x,x)=1·1+1·1 1·0+1·3

0·1+3·1 0·0+3·3=

(2 3

3 9)

The linearityofinner producthasenabled us to writeit as XTAY for

some A.The other twoproperties in the deﬁnitionofan innerproduct tell

us thatAis symmetric,i.e. AT=Aand positive-deﬁnite: XTAX >0for all

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X6=0. Indeed, since (ui,uj)=(uj,ui), Ais symmetric. And the condition

(v,v)>0for v6=0is equivalentto the condition XTAX >0for X6=0.

In fact inner productscan beencoded bypositive-deﬁnite symmetric ma-

trices and every positive-deﬁnite symmetrix matrix Adeﬁnes an innerprod-

uct bythe rule (v,w)=XTAY ,X=[v]A,Y=[w]A.

4Orthogonal bases

Wecanusethe results weknowabout positive-deﬁnite matrices toproduce

some information about inner products. Forinstance wemightask —is

there some basis of Vrelativeto whichthe inner product looks especially

simple?

For this wewill ﬁrst takeitsmatrix Arelativetosome basisA.Since

it is positivedeﬁnite, it can bewritten asBTBfor some invertible matrix

B.Then (v,w)=XTAY =XTBTBY=(BX)T(BY)=˜

X·˜

Y,where

X=[v]A,Y=[w]Aand ˜

X=BX,˜

Y=BY.If wetakeBto bethe

transition matrix to anew basis: B=[I]B,A,then˜

Xand ˜

Yarejust the

coordinates of vand wrelativeto the basis B.

Wehavejust proved thatin some coordinates anyinner product looks

just likethe standard dot-productonRn:

Theorem.Let (−,−)be an inner product on V.Thereexists abasis Bof

Vsuch that (v,w)=˜

X·˜

Y,where˜

X=[v]B,˜

Y=[w]B.

The matrix of (−,−)relative to this basis is the identity matrix 1... 0

.

.

.....

.

.

0... 1.

If Ais any other basis and Ais the matrix of (−,−)relative to A,then

A=BTBwhereB=[I]B,A.

Example:Find abasis Bof R2suchthat the inner product whose matrix

relativetothe standard basis is (2 3

3 9)is given by(v,w)=XTY,where X,Y

arecoordinates of v,wrelativeto B.

Solution:Wecould havetried to write(2 3

3 9)asBTBfor some invertible

matrix B(ﬁrst diagonalize the symmetric matrix (2 3

3 9)in an orthonormal

basis: (2 3

3 9)=PTDP,and then takeB=√DP).

However there is amuchsimpler and moredirect approach: wearetrying

to ﬁnd abasis v1,v2of R2suchthat (v1,v1)=1,(v1,v2)=(v2,v1)=0and

(v2,v2)=1.If the inner product werethe standarddot-product,these would

bethe conditions for ﬁnding an orthonormalbasis. The standardprocedure

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