Question

Corresponding eigenvectors of each eigenvalue 9 Let 2. (as find the eigenvalues of A GA 1 -- 1 and find the or A each 5 Find the corresponding eigenspace to each eigen value of A. Moreover, Find a basis for The Corresponding eigenspace (c) Determine whether A is diagonalizable. If it is, Find a diagonal matrix ) and an invertible matrix P such that p-AP=1

Answer 1

Given 2 0 Az - 0 The eigen eig en values of A is given by the characterestin equation, 13 - 4 x² +59-220 gives, x=1,1,2. compute eigen recles x=1 (A-I ) X=0 NOW where x= У 2 =) :) 13. 2 X-Z - 0 y =) 21Z eigen rector X y Then eigen space Ex={(1) : 27 . 47(5),(3) Now (5),(38) are Linearly independent. Hence clim Ei= 2

Now for X-2 where (A-27)x=0 2 5) - 0 0 x-y-z=0 z=0 = x=y. Hence eigen vector, x= X ( " Eigen space E2 = Ez=1 Spank (!) clem 2+1=3. Since, Site New dim Ei + dim E2 = is linearly independent set of three rectors (300 Mence A is diagonalisable Then there exist an invertible matrin P and Diagonal matrine PAP= D. D such that - Where De () and, P= 0 0 002
There are other arguments aswell by which you can conclude that A is invertible like algebraic multiplicity of each eigen value is equal to the geometric multiplicity. Try to find the other arguments would be helpful for you