Question

Review 4: question 2 Let A be an n x n matrix. Which of the below is/are not true? A Matrix A is diagonalizable if and only if the dimension of each eigenspace is less than the multiplicity of the corresponding eigenvalue. B Matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. Matrix A is diagonalizableif and only if all eigenvalues of A are distinct. D If A is diagonalizable, that is, A = PDP-1, then, Ak = PD*p-1 for k = 1,2..... If A is a 2 x 2 real matrix with non-real imaginary) eigenvalues, then we solve one of the equations of the system (A-1)x= 0 to find a basis for the corresponding eigenspace. E To determine whether the origin in R2 is an attractor, a repeller, or a saddle point of a discrete dynamical system Xx+1 = Axx, we compare with 1 the magnitude of each eigenvalue of A.

Commenting no idea is not helpful and doesn't mean my question needs to be edited. The answer is A and C are false, I'd like a good explanation.

Answer 1

(A) Let A square meitrix Anon dimension of eigen space wilt respect to pigen Valuet is equal to dimension of nato space of (A-d I) dimension af nullspace of (A-d1)= n-PA-dI) ✓ Geometric multiplicity of eigen value G.matd. property A e diagonalisable if and only it Algeboil multiplicity an of eigenvalue'= Geometric multplicity of eigenkul ie A Mord = G. Mark प्र' S Use this property dimension of eigenspace with respect to ergen value's is equal to A Mofi (Algebraic Multiplich tyaf-d) then 'A' is diagonalizable it and only It dimensium as-eigenspace Wilt respect to eigenvaluest & equal to Multiplicitzat the eigensialues' cowegonding given then it is wrong Scanned with CS Cambikin op hun in Less othan

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Simitily D² - party general Ak-pokat d is true Diae Let A be square mutax is A' 23 dicegemodizable then a Non sigulu matrix 'p' such that A=Ppp' • A² (P.DPY) (PDP? PD PPDP (pp = I) =ppept So CamScanner Where k = 1,2--

- Let 'A' is 2x2 seal matry With Nontoro real ergen interes Consider A-fo 2:] => find eigenvales Adl=0 Characterstic equat 'A' is de tracArthro dzo+(-1)=0 = 1+1=0 => d=21 is Nen Real imaginey er genvalues find eigen Vector 1x19th respect to eigen kale dal 4x= [3] → (A-1)x=0 to i -1 1o-1 [41] [3]- Loka->Chate [+][3][] -(1-1=0 t Let y = + (because I'oow independent) x=4 =-t ट * W]/1481 - 1976/768? x = spacey-16 ] } is the Balis of for the Comespending eigen space Yes Itis true CS Scanned with CamScanner