Diagonalization Theorem - Linear Algebra - Quiz Solution, Exercises of Linear Algebra

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¢ Math 205A Quiz 8, page 1 November 30, 2007 NAME cuts aahead Salas 1. Let A € Mixa. We say A is a diagonalizable matrix if and only if there exist two matrices P and D both in Myx. such that: ‘ 4 : 1A. P is what kind of a matrix? indelible. 1B. D is what kind of a matrix? chan onad -l 1C. A equals what product in terms of P and D? POP 65 6 30 5 2.1 A= | -20 3 10 |, then A has eigenvalues 3 and 5. An eigenvector for 3 is | 1 |, and 5 has 120 12 —55 ae multiplicity 2. Use this information and the Diagonalization Theorem to diagonalize the matrix A. (Just find P and D.) we heed te Brod a basis the the eiyen pice f A= 5; ze met hace otnenin 2 ar Grill pe be Hiagonal dalle. Mou heeiytn space Jd 25 ft fe pull sperce J A-SI, bie fad “3 : A-SI = [2 = | ~ e ‘ & => /0x,= “x, +5%, thos Bx, a fs @ fo 1h ~ ol, 4g PR) fntag thx “ye pss fn eee es led génvil A=S 6 UPL [eF o« HF ger Lenk Me a # odgl & ? Le) fe Gay bets D= [: | hun at woh, favee) uill ofp, 3. Give an example of a matrix S € M,. which is invertible, but is not diagonalizable, and explain why S has these two properties. oars “aust iy. eR tal me Aen Oe re ors ‘dep AVO ik how a chirecker hit ph. th, wo teal wot (<0 tha tijtn vies, c no “O") Ar extingle * C 0) 5 dit. 6 Wao Sanh, bt ob cher, py (-a)(i-r) #70 f x = \N2ne/rle ri we Zrell, véhech be An real root, 4, Suppose @ € Mox2 and @ has the form we a | where a is some real numbe' 4A. Find the determinant of Q. det= [@l= ale- (ala) = a)ea'= 0 4B. Is Q invertible? Circle one: Y (x) \* 4C. Find and simplify the characteristic polynomial of Q. polynomial is 1 uM ak a | (wor "*=0") oo = -a->| * z yt 4 a-> = ole wt eS » 4D. Find the eigenvalues of @ along with their multiplicities. eigvals & multiplicities 40 uh id c 4E. Find a basis for the eigenspace of each cigenvalug, most ees preceded We Hic +. with APO, any» a aa aay~ tl K =X, ye -i], ie ae wdexl= (a 2-3 | ae Pee te fee >x nf] 2 Ae ogo 4F. Is Q diagonalizable? Why or why not? Spree f »=0 lee bacis ip. Since. olin 4 Phe 6 ONE bt AZO, ‘Except: Fazoll! Than Os wine! hus matieltihy 25 ve Nor dlirenalieble pmeZ ExCEP TE! aed Q? [32], whch 4 ahiagonalizalle (Gice a aagonat) ant pl Ohen feece 6 Allg Ref m9 DL