extrahw-11-sol

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

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

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

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