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Lecture

# dot product, inner product spaces and coordinates, orthogonal bases, hermitian matrices

This preview shows pages 1-2. to view the full 8 pages of the document. Lecture 13 notes byY. Burda
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,wVso 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.
1
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Only pages 1-2 are available for preview. Some parts have been intentionally blurred. Indeed, linearityand symmetry are easytocheck.Forpositivitynote
that (p(x),p(x)) =p(0)2+p(3)20. 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
.
.
.
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
2
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Only pages 1-2 are available for preview. Some parts have been intentionally blurred. 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-denite 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
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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