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

This is the Quiz Solution of Linear Algebra. Mainly includes points are Explicit Conditionsm, Expansion Across, Equilibrium Prices, Equation, Elementary, Elementary Row etc. Key important points of tags are: Diagonalization Theorem, Information, Diagonalize, Matrix, Eigenvector, Multiplicity, Two Properties, Determinant, Real Number, Characteristic Polynomial

Typology: Exercises

2012/2013

Uploaded on 02/27/2013

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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