5 3continued

EDiagonaliz.lt 1Find invertible P diagonal D such that P AP D

det lA ⼀⼊ I de t l j f I1 ⼋⼆ 1 ⼋ det 只

1 ⼋ ⼀⼋ 2 ⼋

⼋ 1 ⼊z 0 ⼊3 2

For ⼊ 1

1 ii1 iiii炎岩 0 ⼆HI is an

eigenutcowes.to⼋

X了 free

For ⼋ ⼆0

0

1 i 11 i 11 i166 iio o o o

1 i䧔燕 叫⻔ 品 吢恐

For ⼊3 2

1 i 11 器 1

1 i㡭 i 1

Hi it 品品 答

叭舆 叫 in

P A P D i i ii Ai 5

Use diagonalizafion to compute power of matrix

If P A P D A PDP

Ah PDPIPODMO r OPDMPODP iPDF.ktin.esD 1 D D Iii 川

区 Compute A5 for pains A 1P Hi D f 02To get P if 吃5251

Iii i 别ioilòiǐiòǐ

1

iiiiidll.ioo o

ioii.it11

1 0 1 0 0

8 ii i11

1 0 i 1 0 00 1 0 1 1 0 10 i

l ii p i i y i 川 ⻔⽚

1 器点1

iii ⻔______

ㄒ上 let A bean nxn matrix w distinct

eigenvalues ⼋ ⼀

ippsn.laDenote h as the eigencpacecorres to.lu Then

dim lUn E multiplicity of MeL as in char poly

P

lb A is diagonalitable if f Èdinlh n

i ff det A ⼋I canbefactored into linear factors and

holds in a for every h

I C l In the case of lb let Bnbe a basis of hThen B U_U Bp forus an eigenbasis

Suppose detlA⼀⼊ I ⼊⼀⼋2⼋⼀⼊2

了 n 5dimlV.IE 2 di n l h E 3din M di l h E 5 is always true

Show that A 19 6 is not diagonal

izable.det lA⼀⼊ I det f N ⼀⼊11⼋ 1 灬

N 1

It cannot be factorized into linear factors using realnumbers

Therefore A is not diagonalizable

Show that A 19 I is not diagonal

izable.det lA ⼀⼊ I det H ⼭ ⼀⼊ 2 ⼊ 1 ⼀⼋⼆ ⼊2 2⼊ 1 灬 i

N 1 multiplicity 2

For ⼋⼆

I u ⼼ A n I

问 i1 fre variable

dim Y I 2

Therefore A is not diago

alizahe.EEDiagonal ize A 信⾔

det A ⼀⼊ I 12 ⼋ 1 1 ⼋ 12 ⼊ 2 ⼋2

1 ⼋

⼋ 2 1multiplicity 2 ⼊2 1 1multiplicity

For ⼊ 2

1 i 11 i 110 ⼝ o o

o o o o

ii

o jh 0 din V 2

X3 freelequal to multiplicityof ⼋

i 1 川 烨1 倒了 is a basis of u

For ⼋ 1

1 iiii㘜 州 1㖄

l f is a basis of Nz

Twehe.lk 191 If is a basis of 1123

on尛⼊ 㕾 ⼊Forres to ⼋

叫 D 12 2 TAP D

叫 6 年1 5 12 121