extrahw-11-sol

extrahw-11-sol - MATH 108 B FALL 2011 EXTRAFCREDIT PROBLEMS...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 108 B FALL 2011 EXTRAFCREDIT PROBLEMS SHOW YOUR WORK CLEARLY. ' *1) Let A be a n x 'n, skew~Hermitian matrix (A* = ET —A). Prove: (a) T he eigenvalues of A are pure imaginary, and eigenvectors corresponding to difierent eigenvalues are orthogonal. {we recall that if m, y E C“, then (93,9!) : mTy : 221:1 33' w“ J H. m5 1 )4x,x>z¢>g,x>: 3H9 J z» .5” (‘3? _? 5H” F“ "A???" <¥/&>X\?z Kl > .:€A§f>:4x,5x7o # to: y M (KW-F5 # x go >\ I k E: “wisp {‘7 6"}: fi-kkpg w flag“? ":7 flaw“ —-‘-" )rZ’sz‘l—7Z {Aha/£27 “$2 (b) Prove that (I A A) and (I + A) are invertible. lax—Q‘s} mkwflmofibfi _ ® a7 (Rib: D. “3%. E; awwxfin A k {ammkbm 'l Q‘s - * i kwgihwvx ’r' Mfik :1 I 4}: 'x 7' MIX” m 'W’miMM “(‘3 A i ' X 5r {3: aemmxae::fiw&“f,“ “13‘ (6*‘>¢(2;%B= = 6‘ age}: ékflfiv 63%3- gd) Q : (I _ A)(I+A)—1 is an 3%, é um (Kw) (Ha) ( M)" figfsékfy CH“? S’P‘MHKY: (yvm‘fimfi compute Q as impart (g). 33- :x 2 (\‘z ’3} X»)ch :: \ » +51 \v-Z. g\ i 2 p2 ‘ > (2) $4; M ) U"M Wfi. @ Q+ l9}: Xflm‘ MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS *2) If the vectors m1 and a: eigenvectors of ? 5w :5 ‘ &L. 9:) :3 $7 gawk-am; QEfils’gL R EEEEEN %§L $®&%€fifi ffiikgk& §fl 5:); 3 2 are the columns of S, what are the eigenvalues and r 23 _1 c_s<01)s ? 3 BwfiEflfi ‘26} 2% <3 x \i! w 2:.» fiw'wmrl? “$3? 4 -3) S'WEGf—ue 1%,; v??? = 09f)“ 4 wad ‘1 (b) éfibg MATH 108 B FALL 2011 EXTRA—CREDIT PROBLEMS E C is an eigenvalue of B, then W = 1. 89 (a) Show that if B is unitary and A $§a>i. he W7 r? * r2 "M? Q figmax Show that if A is normal (1.1a. A*A = AA*) and invertible, rhen B : A*A— ‘ is unitary. 15% e W W E K‘ 1 (NM (M I «:0 <39? M fit) X-Mflm 1 -r‘é 1;};4'F “‘ A>f¥ AA *KWQX‘Y‘NMWL $9“ MATH 108 B FALL 2011 EXTRA-CREDIT PROBLEMS 5 A:(3 has no square root, Le. there is no B such that 32 = A. 4) Show that B:- (62:33} MQLM E3: (é 3>Cfic:> 1 is” t ‘ g. Q? 'Qabcfl O of“; Joe: —: aéi-CcLzo f7 \C(a¥é§>':g mom/3.9.3- Lag {Jéf’fC} *WWM ...
View Full Document

This note was uploaded on 12/26/2011 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.

Page1 / 5

extrahw-11-sol - MATH 108 B FALL 2011 EXTRAFCREDIT PROBLEMS...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online