Today § 6.1/6.7 : Inner Products

Next : § 6.2 : Orthogonal . y

Reminders

My Math Lab Homework # 7 : Due THURSDAY by ltispm

MATLAB Homework # 5 : Due FRIDAY by 11:59pm

Final Exam is on Saturday,

March '7.ly?foapmm

@ the BEGINNING of Exam.

Week.

An nxn matrix A is diagonal sizable if A =

PBPtenvertiblT

diagonal

Thisis equivalent to Rn having a basis of eigenvectors for A.

This happens , e.g.,

wkw A has n distinct real eigen values.

But it cay fail .

↳ Eigen values : roots of PACD = detllt - XI ) ← degree n polynomialmust have all real roots for Ato be diagonal izable

.

↳ The algebraic multiplicity of X is the degree of this root in p*The geometric multiplicityof X is dim . Nul ( A- XI )

.

Always havegeom.mu It A) falg.mult.ly an be §

A= Hosey Bs HeIII. Pa

'tPBMaIg§¥f!YYsz

dim Nul ( A - SI ) = 2,

dim Null B- SI ) . -1.

" 4 ' s ±

Theorem If A has a repeated eigenvatue X,

then

dim Nul ( A- II ) s multiplicity of X.

The matrix is diagonal izable if and only if

PA has only real roots,

and dim dull A- II )= multiplicity of X for each X

.

In this case , any bases for the eigenspaces

together form a basis for R"

.

So,

just whizb matrices are diagonal izable ?That's a really tough question to answer

.

A partial answer

will be given at the end of the course.

To get there.

..

§ 6.1 Definition : The dot product or inner producton 112

"

is defined by it nxn

m -

y . .V= < b ,y > = u.tv = yty ¥ butnxl hxl lxh NX ) ↳ nxllxw

ftp./Ygf=luikujuH=uii+kv+usg .

Properties:c

, a.v. =±u .

fold(2) ( y,

+ b ) . .v= bit tuz . I }tissaysif {= uu ;D

1- ( g)' - g.v. We,±u.y±= l.u.tt#v=.nPv+ufIvT : R "→1R is

almewtrarsfomatn (3) ¢y) . .v

(4) b.y= up +422+45>0 unless be, hwkhjytf

.

The length of a vector is defined t.be

11×11 = VIMore generally

,the distance between two vectors y

,!

13 distly,

!)= Hu.

- tll x,

E.g. a. =H| ,

tffl:

2.

,- - - - . . .

,

t ;

drstlu .u=Hb-±H=H # H -

ax .

/ / l / Xz-

.

y

=jM4C -25+1-25.

=jTt4t=r=3.

.

X,

How does11A. v. 11 relate to Hull,

11111 ?

:-1112 = ( b- I, y- v. )

= < ±,u . ± ) + s - ± ,u.

. x > -

=ky,u.

) +4¥'+t±u .> +2¥ - t >

= < is ↳ > - < a ,v.

> - < %u. > + tiny >

= 116112 - ZCY,

.V > + 111112

i. 114112+111112--1141112+2<4,

! >

If < apses then utrrectsntem to Pythagoras.

NYH 4118112 . My. V. 112

*If it's o

,we get JTkorm

,✓

u.tv applies to rghtagks .

Law of cosines In 2 dimensions ) :

bC

a2tb2=c2+2abces€a) a=HbH b=lKH c=Hy .gl

" (nylptnvlp .tn#p+2nulllKll6soat

; Yas " n' + " IH'

' Hy-±Nt2( y,

, >

4< y, .v ) = 1141111.41656 .

< a ,v.

>Gsb E Ebl ] 8=65'( pay , )

o÷vs-

-

-

Cauchy - Schwarz Inequalityt

Riesz ly. v. I s 1141111111.

Hike → Segal → Gross → Kemp

Pf. fct )= in-tvlp 30

= s a- t.v.u.tv > = < u. ,u,

> - tcu ,D - Kyu.

>

+

ttkv.us#.:meIy*Et*" it

nots :- b±IbT4ao .

:b'

. 4ac.co

← Hu,±>Tf 411411411112