Homework SourseFor the following matrix a find the characteristic equation
For the following matrix:
(a) find the characteristic equation
(b) find all eigenvalues
(c) find all eigenvectors corresponding to each eigenvalue
(d) choose as many linearly independent eigenvectors as possible for each eigenvalue.
[1 1 1
2 1 -1
0 -1 1]
Solution
Let the matrix be represented by A
a) In order to find the charateristic Equation,
det (A - I)= 0
det 1- 1 1
2 1- -1 = 0
0 -1 1-
(1-) [(1-2)-1] - 1 (2-2) + (-2) = 0
(1-) ( + 2 -2-1) - 2 +2+2=0
(1-)(-2)+2=0
2 -3 +22 =0
-3 +32=0 This is the characteristic equation
b) By solving above equation
2 ( - + 3 ) = 0
Therefore = 3, 0 ,0 are the eigen values
c) & d) Now the eigen vector for =3
[ -2 1 1 [x1 [ 0
2 -2 -1 x2 = 0
0 -1 2 ] x3] 0]
let us take the augmented matrix
[ -2 1 1 0
2 -2 -1 0
0 -1 -2 0 ]
R3 = -R1+R2
R2= R1+R2
[ -2 1 1 0
0 -1 0 0
0 0 -2 0 ]
R1=R1+R2
[ -2 0 1 0
0 -1 0 0
0 0 -2 0 ]
Therefore -2X1+X3=0
-X2=0
-2X3=0
Therefore X1= X2=X3=0
Eigen vector= [ 0]
Hence this chaacterstic equation has got a single solution
For = 0
[ 1 1 1 [ x1 [ 0
2 1 -1 x2 = 0
0 -1 1 ] x3] 0 ]
let us take the augmented matrix
[ 1 1 1 0
2 1 -1 0
0 -1 1 0 ]
R3=R2+R3
R2=R1+R2
[ 1 1 1 0
3 2 0 0
2 0 0 0 ]
x1+x2+x3=0
3x1+2x2=0
2x1=0
Threfore x1 = x2= x3=0
Theeigen vector will be [ 0
0
0 ]
Even this eigen vector has got a single solution
