Question

Let the matrix below act on C? Find the eigenvalues and a basis for each eigenspace in c? 1 2 - 2 1 1 2 The eigenvalues of - 2 1 (Type an exact answer, using radicals and i as needed. Use a comma to separate answers as needed.) are A basis for the eigenspace corresponding to the eigenvalue a + bi, where b>0, is (Type an exact answer, using radicals and i as needed.) A basis for the eigenspace corresponding to the eigenvalue a - bi where b>0, is (Type an exact answer, using radicals and i as needed.)

Answer 1

SOLUTIONS 2 Let A= [] Its eigen value equation IA-Il=0 |-入 2 1-1 10 - 2 7 1. В (1-2) 2 +9=0 to (1-1)² = -1 1-221 ,= 1tdi, Hence Eigen values 14,=1+2i, t2 = 1-2i =) d2=1-2i are Eigen vector corresponding t;= 1+2i [A-1, ] x=0 =) 1-(1+2i) 2 -2 1-(1+2i) 7631-13) 32] [3]-[:] a 2 =) - 2 R2② R₂+iR, -zi 17 2] []-[:]

Here X2 is free variable assign x2=1 - 2 i xit 2x2 = 0 a 25 - 다. Hence X, - [1] is the basis of eigen space for ti Eigen vector corresponding t2 = 1-2i [A-da []x=0 1-C1-2i) 1- 1-(1-2i J[*}]: [] :][:] -16) di di Re R2-ir, di Here x is free variable assign X2=0 aire et 2X2 = 0 ľ +22=0 x = -11 x 1 / 4 = i Hence X₂= [:] is the basis of eigen space for a

Boxes portions are final answers respectively