M294F09WS_EigenvaluesSol

# M294F09WS_EigenvaluesSol - Math 2940 Worksheet: Ch. 7...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 2940 Worksheet: Ch. 7 Eigenvalues and Eigenvectors November 5, 2009 (Q9 /tt74mg 1. Find all eigenvalues, with their multiplicities, for the matrix Zwi 0 C) C) fir/if": / ’71 23/, ff ] Efﬁmr-tmw' z. 2 I 2, 1* A [ONION l—‘l—‘l—lo NNOO r» W M “M? W )('/~vl)(2r~~i){z.l) : [Z'~1)1//~xl ﬁg? Wale/15L 2. Find the eigenvalues and eigenvectors of the matrix 5 l7 . Mr : [2"1 if", a MMair):/s~:,1>/v/~w2)m) will/Awlhyrﬂkwfxﬂlf14/357 MM“, 1,, ﬁg»: 17” a mat—Van :9 W» 1.722272) ﬂimsy Maw/W) ‘ ‘ ~ . - " 7.“ Wuth my tarrﬁngﬂim a, aljm, WJ will] n’? t a, v _ 4v w m W 52,. 4791472 52: Ariqu 3—) #7417 a? a»? @P-ﬂﬂ)? u , v ~ " #312“:ng :7, p,;,?][,:7,[/,[§7% Mt w M 3. If A is a 2 X 2 matrix with trA = 1 and detA = —12, What are the eigenva ues o varying Ma/meé’mﬂc ﬂ/MMg/y ﬂaw/ﬁﬂMVLMﬂ ‘ It WM: 2 "-~ xiv/K aha/[Wm MW, (/zn- 4! W «3] 4. Suppose matrix A is similar to matrix B, (that is, AS = SB for some invertible matrix S, or equivalently, A = SBSJ). What is the relationship betWeen the eigenvalues of A and B? H b tth ' t 7 AL, I 1 ﬂ ’Mb :11? jug ZelfSyiijSM/W mg V ' ﬂ 7 47.09%! VF Avvirsi, 99 1’65”? all?” a Mgr“: SW =>I Eé‘""v”9= l/JriVL eigenvector '17. Is ’3 an eigenvector for A’l? If it is, what is the corresponding eigenvalue? MW e ﬂiﬂvsﬂr/2v2e 721%"? af7;#”’7 6. Let A b as abov. Is 17 an eigevector i0 A5,?‘I‘f so, what is Rte associated eigenvalue? W .7 s M [email protected]) erg/thg/rf/ﬂgg/Q Mf/V r 5127 ,p" trademark" or ,, Wéz/r?‘ 7. Arguing geometrically, ﬁnd all eigenvectors and eigenvalues of the linear transormations given. If possible, ﬁnd a basis of 1R” consisting of eigenvectors. V r, ,. - “A 7 ﬂ/' / I“ 7%, 9 ‘l (b) Orthogonal proje “7/21? 197117 v), A? v Na 7, if (an/Ml /0 L, ,ﬂfrz 2:0 /=% a] i (“LEW/aways, L Mmp’canwééyé r/ g: 8. If 0 is an eigenvalue of a matrix A, how many linearly independent eigenvectors oes 55 so 7KW¢7MA4£MM¢ gal/a {ﬁr/4) 7 9, Find all 2 X 2 matrices for which 61 is an eigenvector, Is the set of all such matrices a subspace of the set of all 2 x 2 matrices? If [it is, ﬁnd a basis. jgfoﬂj W [if W Z M/Wﬂ’ #1 things; their only earthly possession is a small purse with a bit of gold dust. Each day they get together for the following bizarre bonding ritual: Each of them takes his purse and gives his gold away to thevtwo others, in equal parts. For example, if Abraham has 4 ounces one day, he will give 2 each m Benjamin and Chaim lit? a /M MMW a) If Abraham starts out with 6 ounces, Benjamin with 1 ounce, and Chaim with 2 ounces, ﬁnd formulas for the amounts a(t), b(t), and 005) each will have after t distributions. 1 1 1 . w 6‘ (Hint: the vectors [1], [0 ] , and [—1] will be useful) W .. 1 —1 0 a i ‘9 " ‘77 a V; A a Hail/#567 (b) Who will have the ost gold after one year, t t is, after 365 distributions? f 1/ 2 b{'{)+ icy): 4/19” ' -a/f/l . aNW _» ‘ W 73% - J a) + am 3 MW) W i W”) W Maw M f (1/!) J; l [m] a”) m“) + s c . z a a (1‘ “it? . 3 j t. r if. agar édeVZ/gj . ,A WréﬁW : bf m ‘ _ __. j G (g) 9% 7/0: Z jag/l, 706%) Z ""0 .1 a 0 A J 460‘ .21 /MI) 5 (' 4‘ ‘21 0 _. W m ,4 (74~/%v/o//2) :ﬂ/ﬂv/ll/ , / . ML .: Vzﬂf) ‘7 g ﬁg M144” V/ﬂ‘r’zojf/ m jW ‘ ‘ ,, *i ,I/ /‘ o‘/'/2 I .://' 7 - 5 z .- x a .2 J; We 0 , [Mg/q [/2 v (:2, l “ II / ya” 0 ~( y,» 2 4/ Vlt/L'o 0 o ‘ L z I > . m» ,_.. ﬂ. ﬂ “Km-M ﬂﬁwsv? , y , W /#.1 .-./_ ya ‘7? ,g; W¢Wﬂm mwymﬂ/ﬁm WW 4’ 2 2/ 2,. ( q "’ I “I /‘ ‘~> "5, ~f> , L. / «,4 a v 570 mm w {9, M 47/74"; 992W“? //“(/{VZg/j 5%? L ) 2/ . {Va LC Mia; / awe/HM, ﬂy M M259 cm ﬂr/‘yzwr 9% ~W/ Ag [w WW )0 Lama/K, 34‘ ...
View Full Document

## This note was uploaded on 12/25/2009 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell.

### Page1 / 4

M294F09WS_EigenvaluesSol - Math 2940 Worksheet: Ch. 7...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online