Question

Consider matrices 2 A= 2 -4 1 and -8 12 -2 3

Find a basis for the eigenspaces of matrix A.
What is the algebraic and geometric multiplicities of its eigenvalues.

Answer 1

Q. Here - 1 2 A 2 -4 D The charecteristic equation of A | A-XII=0 -|-) 2 2 -(1+x) 2 2 -(4+7) -4-2 (1+) (4+) - 4 - 0 * 4 +7+42+72-4=0 * x2 + 5x = 0 * x(x+5)=0 So, x=0 x+5=0 1-5 so, >,=0. and X2=-5 be the eigen value ob A. Now for x = 0, (A-X, I)x=0 (A - OI)X = 0 -1 2 1X=0 2 -4 NI here X = N2 - 1 2 ni So, 0 N2 2

-n, +202=0 = n,- 202 = 0 2n, - 412 = 0 ,- 202 = 0 2 So, from 2 (2) n, = 202 here na = a then n=2a. So, Х a a So,v= xt? be the eigen vector for >,=0 For, A2=-5 (A-82 ) x = 0 + A+5]) X = 0 -115 2 2 -4+5 ni 1110 2 ni 0 4 2 0 12 So, unit 222=0 - 20 th 2 = 0 - 1 2nit 12=0 © & (i are same. so 2n, th₂=0 N₂=-2n, Let So, n₂ =- 2b. n = b b 1 ni n2 - 2b

So, 2 1 -2 be the eigen vector for 12= -5 for a value. Algraje multiplicity - Number of multiplicity eigen Geometric multiplicity Number of eigen vector bor corresponding eigen value So Answer: Basis for the eigen spaces of matrix A is No, V2} - 2 1 2 for x = 0 Algebroic multiplicity. [1] geometric mulliplicity -[1] for 72 = -5 Algebraic multiplicity 1 Geometric multiplicity = 1
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