Today : § 3.3 Determinants

&§5.1 : Eigen values

Next : § 5.2 : Characteristic Polynomial

Reminders

MATLAB Assignment # 4.Due TONIGHT by 11:59pm

My Math Lab Homework # 6 : Due Tuesday by ltisepm

Midterm # 2 : Next Wednesday,

8- 10pmRom 1 seat assgn .

& practicemidterms posted on course webpage .

Properties of Determinants

* Defined by cofactor expansion ( recursive expression , expandalong any now or column )

* Behaves nicely under new ops : detA= ± ( product of pivots inrow reduction Kerrey ) ) .

* detA=o iff A is net Invertible* det : Mn → IR is multi linear in each now and column of

the matrix,

with the others held fixed .

* dot LAB) = detlA)deHB ).

.: detl An)=(deHA)Y .

Two MORE PROPERTIES :

det l At ) = Ydef Adot ( cadb )

= ad - be

"

det l At ) = det (A) debtgcd)

= ad - Cb

What is the determinant ? What does it mean ?

Parallelograms :

b.b.to

S

A IQ'

g

Area Plaobj + Area Pla. ,s| = Area P[ a

,

bts ]

: the R is multi linear

d La ,b I + d [ a.,

s ) = d ( a. b.ts ).

heorem . Given two vectors g. b. e- 1122,

the area ofthe parallelogram they determine B

Ala. ,

b.) = ldetla,b.)

lydettdettt" '* "

tutaidwhat dees the sign of detlg

,bt mean ?

orientation.

What about nxn determinants with n > 2 ?

Parallelepipeds taped

p Idetlabcjfldumelp )

§ 5.1 / A linear transformation T :v→V tends to meue

vectors around.

Eg . TIM

.tn#=l3oylgsTl!2)=sojogenvawe412,1t.ly/=ygyoe9enEK

Eg .

If A is a stochastic matrix ( new sums =| ) thene%Yt↳

De¥t¥¥tgIdH=tgt¥eh¥¥M.Hi .tt#- eigenvector

Definition Lett :V→V be a linear transformation .

If there B a non - zero I← V such that

Tt@YYef.rs.mIsYalaiFeR.wecall v. an

eigenvector of T.

The scalar X is

the corresponding eigen value.

( We similarly

talk about eigenvectors trgenvalnes ofsquare

matrices.)

Eg.

Show that 7 is an eigenvalueof A =L 's{ftp.fyjffpg)

Ie.

there 's Its with AI ' 7k.

n =7H )i.e. e. = A .x

- Zx = (A - 7I ) I want

a # Halted 'f8fH 's:H→HHattabaugh

Theorem : Forany he R

,

the set of vectors v. for whrh

TH ) = tv

B a subspace of V,

called the eigenspace of X.( ( We only call d an eyen value if its eigenspale B He})

{ ± : Tkthv } = Nul ( T . JI ) Is a subspace .

Hit could be {e. } . If so

,X

B ne± an eiyenuahre .

Eg.

The rotation matrix ( o

, f) kcwsi ) has no

eizenvahei.

If d were an egenvahe ,

then the is an egnvectr HIT feel

"

t¥Y¥:*,µ .:¥%Y¥a-

y .

. Hy i. HIDY- o

> o; .y=@(¥:@

* Given a ( potential ) eigen value,

finding the corresponding

eigenspaee is noutme.

* Finding the eigen values is typically hard. ( Row reduction

changes the eigenvakes . )