For each of the following matrices find its eigenvalues eige

For each of the following matrices find its eigenvalues, eigenvectors, and determine if it is diagonalizable; in which case find P such that A = PDP^-1. (a) [1 6 0 -1] (b) [3 0 2 3] (c) [2 1 -1 4] (d) [1 4 3 2] (e) [0 2 3 1 1 3 1 2 2] (f) [3 1 1 1 3 1 1 1 3] (g) [2 1 -1 2 3 -2 -1 -1 2] (h) [2 1 0 0 3 0 -2 2 3] (i) [0 1 -1 -1 2 -1 -1 1 0] (j) [1 2 1 2 5 3 -3 -2 1] (k) [2 2 2 0 2 2 0 0 2] (l) [2 3 2 -2 -3 -2 -2 -2 -2] (m) [5 0 0 0 -3 3 0 0 0 1 2 0 9 -2 0 2] (n) [3 0 0 1 0 2 0 0 0 0 2 0 0 0 0 3]

Solution

(i) eigenvalues ₁ =1( of multiplicity 2), and ₂ = 0. Corresponding eigenvectors v₁= (-1,0,1)^T ,v₂ = (-1,1,0)^T and v₃ =(1,-1,1)^T . The matrix is diagonalizable. P =

-1

-1

1

0

1

-1

1

0

1

(j) eigenvalues ₁ =1/2(7+i15), ₂ =1/2(7-i15), and ₃ =0. Corresponding eigenvectors v₁= (i/8(i+15),1/8(7+i15,1)^T ,v₂ = (-i/8(-i+15),1/8(7-i15,1)^T and v₃ =(11,-4,1)^T . The matrix is diagonalizable. P is the matrix with v₁,v₂,v₃ as columns.

(k) eigenvalues ₁ =2( of multiplicity 3). Corresponding eigenvectors v₁=(0,0,1)^T. The matrix is not diagonalizable.

(l) eigenvalues ₁ =-2, ₂ =-1 and ₃ =0. Corresponding eigenvectors v₁= (1,1,1)^T ,v₂ = (2,1,2)^T and v₃ =(1,1,0)^T . The matrix is diagonalizable. P =

1

2

1

1

1

1

1

2

0

(m) eigenvalues ₁ =5, ₂ =3, ₃ =2(of multiplicity2). Corresponding eigenvectors v₁= (1,0,0,0)^T , v₂ =(3,2,0,0)^T ,v₃ =(-1,2,0,1)^T and v₄ =(-1,-1,1,0)^T . The matrix is diagonalizable. P is the matrix with v₁,v₂,v₃,v₄ as columns.

(n) eigenvalues ₁ =3 and ₂ =2 ( of multiplicity 2 each) . Corresponding eigenvectors v₁= (0,0,0,1)^T (correspnoding to ₁ =3), v₂ =(0,0,1,0)^T and v₃ =(0,1,0,0)^T(correspnoding to ₂ =2) . The matrix is not diagonalizable.

    

-1	-1	1
0	1	-1
1	0	1