Question

7.1.21 Question Help Orthogonally diagonalize the matrix, giving an orthogonal matrix and a diagonal matrix D. To save time, the eigenvalues are 17, 13, and 1. 8 7 1 1 Enter the matrices P and D below. 0 0 22 2 3 0 0 1 0 0 0 0 1 0 0 0 0 13 0 0 0 0 17 - 1 1 1 (Use a comma to separate matrices as needed. Type exact answers, using radicals as needed. Do not label the matrices.)

Answer 1

Aiagematial iniasly ndfenduqvetes A iagmaliable Hhere matux A iA An it and And values f A n. (wot necessarily distinet) and , ,- O 0 PAP D and P and D i with A =PDF Scanned with CS CamScanner

A - L envalues ar 113, and i . gen nee in я. irace A 8+888 17+ 13+ 31 + with mulkiplscty A is ain value gen values are 13 veters willndgem igen vecte A nen-0 veter (A-AI) Scanned with CS CamScanner

(A-A.I) Cosidur 8-1 8-1 8-Vit 1 whugmemked Malin in 0 0 7 1 1 0 4 48 0 R37R3-R 48 48 R4 R4-R 7 0 48 0 48 Scanned with CS CamScanner O O O оо O

Pivet elemont he t element Зoно whicls belos in that all elementaane o columninalled Pvol element Pevat Column Cblumn having pivt elment Pivot columm called in Non - olumna oreafeonds ivet variabe CS Scanned with CamScanner

st and 3rd eolumn have bivot. Here ad and 4th eolumn in Non- bivot column. variables . are fre and Vi4 12 variables As we have two frce indep endant eizenvectes set tin tinearig we Eirat takı : 48 Vi3 + 48 V14 : %3D A 48 Vi3 + 0 = 0 U13 7 Viz +7 V2+ Viz + Vị4 = 0 7 Vu + 7 xI + 0 + 0 = 0 dondly take , Viz = 0 ; Vi4 = 1 48 Vi3 t 48 Äl =0 cs Scanned with CamScanner

V 0 O - 13 8-13 T 7 8-13 33 8-13 8-13 5 Madrino in -5 0 -5 t -5 1 0 3451512s0 24 360 36 -34 0 0 caa CamScanner -

L 5 1250 3415 30 0 -30 30-30 0 RA 34/5 s 12/50 -30 36 O 4th columm in Non-pivet column dakuV3t - 30 v33 3 oV34 t 0 + + 1 aV3 - 5 3 -5 =0 Scanned with CS CamScanner

Cemaida (A-I 17 41 |V42 0 El-8 8-17 0 8-17 8-1744 -9 7 vhugmanted Mahin 0 -9 -9 -9 10 -9 L 16 80 640 64 -80 0 16 R43RRI 0 о -з3у3 145 o -72 72 0 72-120 0 0 1 0 16/ 16/50 0 -3/9 R4R4 t3 6 Scanned with CS CamScanner

on-pivst vanabu ake 44 3 V43 ay 3 a 7 = 0 72 V43 43 16a 4 ynd -31 4 2 16 16 142 -91 4 2 0 CS CamScanner Scanned with

Construct P= Lui Vg Vg VAl -1 -- 1 and P %3D dz 13 dz da 17 p and D i this with p' AP AT - Wow %3D as 8. : A in symmetric matrin And, eigenivetos a are outhogonal นี, ปี น. ี้ symmetic motim eutuoganal evéhozonal matri. A is eithogonally hiaganaliable. P is an and, hence Scanned CS CamScanner

Ortho ragonally gnalae: A mati A evikojenallyaonaath ihayonal matin pAP (dagaal mabn he eaist an P Auch that tgoaly agonaleabe A with P = - -L -1 0 and D = O 0 3 17 Cs Scanned with CamScanner