Question

Consider the 2×22×2 matrix AA given by A=1−2[−5−1−1−5].A=1−2[−5−1−1−5].. Find the eigenvalues λ+λ+ and λ−λ−, larger and smaller or equal or conjugate, respectively, of the matrix AA,

I am really stuck on parts b and c so any help would be greatly appreciated!

(10 points) 5 Consider the 2 x 2 matrix A given by A al -}] 1 a. (2/10) Find the eigenvalues l_ and _, larger and smaller or equal or conjugate, respectively, of the matrix A, + = 3 Σ X = 2 Σ b. (4/10) Find corresponding eigenvectors vt and v, such that vt has component vi =1, and similarly for v. y+ = M V M C. (4/10) Find the corresponding unit eigenvectors ït and û , such that ởt has component ✓ 1 > 0, and similarly for . + = M v = M

Answer 1

The given matrix is 5² Y₂ Az 155's] Y 5/2 lues we consider the equation Tobind the exgenvalu 5%-7 YZ Y2 5-7 FO =0 5 2 m om ->)² + 4 25 - 54+x=0 2²-57+6=0 * = 3x-2x+6=0 ^(9-3)-2 (9-3) = 0-3)(2-2) m n = 2 either a=302=2 There for At=3 3], a. Now we find the eigen vector for & 3 5-3 Y2 Y 2 - 3 Then -3][23]-[:]

3][3] [:] 17 - + - 이 개적이 0) 시 2 Thereture from above and equation 시면S), FR Therefore the eigen vector corresponding to x=3 is ) for KER now for x=2 K Then we have - 2 틀 브 로들 )() (8) - 2 . 000 0 초월 초초 0 ( (2) - 1 찢어를 쫄를 Solve these two equation weget 개로 det rez=K, ER any number) Then 매다 Therefore the eigen rector corresponding to x=2 rs v=ki(7), KER (6). Therefore vt cosides - (1) (K) be any real number (C) The corresponding unit eigen reeters Vt 옩 솔 을 옳 - : () = 1 )