EigenVector.md (1485B)
1 # Eigen Vector 2 3 Self Study 4 5 **Definition:** An Eigen Vector is a non-zero vector that when a linear transformation is performed upon it, the resulting vector is only moved by a scalar multiple (remains on the same line). 6 7 Associated with this, we also have an Eigen value which is the amount that a point on the Eigen Vector is distorted by (multiplied by this scalar) 8 9 This can be thought of as the axis of rotation when in R^3. Additionally, we know the eigen value is 1 in a rotation as there is no stretching. 10 11 Formula: 12 13 Where T(x) is a L.T., A is L.T.'s matrix, v is a vector, lambda is a scalar 14 15 T(v) = Av = lambda v 16 17 There are eigen vectors iff det(lambda I_n - A) = 0 18 19 ## Calculation 20 21 A = [1 2] 22 [4 3] 23 24 Eigen Value Calculation: 25 26 det(lambda [1 0] - [1 2]) = 0 27 [0 1] [4 3] 28 29 det([lambda 0] - [1 2]) = 0 30 [0 lambda] [4 3] 31 32 det([lamda-1 -2]) = 0 33 [-4 lambda-3] 34 35 (lambda-1) (lambda-3) - 8 = 0 36 37 lambda^2 - 4lambda - 5 = 0 38 39 (lambda-5)(lambda + 1) = 0 40 41 Solutions: 42 43 lambda = 5 or lambda = -1 44 45 (lambda = eigen value) 46 47 Eigen Vector Calculation (calculating for 5): 48 49 0 = (lambda I_n - A) v 50 51 = ([5 0] - [1 2] ) v 52 [0 5] [4 3] 53 54 = [4 -2] v 55 [-4 2] 56 57 Null space calculation: 58 59 [4 -2] 60 [-4 2] 61 62 [4 -2] 63 [0 0] 64 65 [1 -1/2][v_1] = [0] 66 [0 0][v_2] [0] 67 68 v_1 - 1/2v_2 = 0 69 70 v_1 = 1/2 v_2 71 72 E_5 + {[v_1] = t[1/2], t \in \R} 73 [v_2] [1 ] 74 75 Answer: 76 span([1/2]) 77 [ 1] 78 79 The other calculation would be the same but for -1 thus it will be left as an exercise for the reader.