eigenvalue matrix 2x2
code
import math
import numpy as np
from numpy.linalg import eig
# matrix
print("M = ")
M = np.array([
[2, 9],
[8, 1]
])
print(M)
print()
# eigenvalue, eigenvector
val, vec = eig(M)
# eigenvalues
print("Eigenvalue =", val)
print()
# eigenvectors
print("v =")
print(vec)
print()
v1 = vec[:, 0]
print("v1 =", v1)
print()
v2 = vec[:, 1]
print("v2 =", v2)
print()
# check relations
Mv1 = M @ v1
print("M v1 / val1 =", end=' ')
print(Mv1 / val[0])
print()
Mv2 = M @ v2
print("M v2 / val2 =", end=' ')
print(Mv2 / val[1])
print()
results
M =
[[2 9]
[8 1]]
Eigenvalue = [10. -7.]
v =
[[ 0.74740932 -0.70710678]
[ 0.66436384 0.70710678]]
v1 = [0.74740932 0.66436384]
v2 = [-0.70710678 0.70710678]
M v1 / val1 = [0.74740932 0.66436384]
M v2 / val2 = [-0.70710678 0.70710678]
theory
- matrix $$\tag{1} \left[ \begin{matrix} 2 & 9 \newline 8 & 1 \end{matrix} \right] \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] = \lambda \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] $$
$$\tag{2} \left[ \begin{matrix} 2 & 9 \newline 8 & 1 \end{matrix} \right] \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] = \lambda \left[ \begin{matrix} 1 & 0 \newline 0 & 1 \end{matrix} \right] \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] $$
$$\tag{3} \left[ \begin{matrix} 2 & 9 \newline 8 & 1 \end{matrix} \right] \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] - \lambda \left[ \begin{matrix} 1 & 0 \newline 0 & 1 \end{matrix} \right] \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] = 0 $$
$$\tag{4} \left[ \begin{matrix} 2 - \lambda & 9 \newline 8 & 1 - \lambda \end{matrix} \right] \left[ \begin{matrix} v_1 \newline v_2 \end{matrix} \right] = 0 $$
- determinant $|M| = 0$
$$\tag{5} \begin{array}{r} (2 - \lambda)(1 - \lambda) - 9 \cdot 8 = 0 \newline 2\cdot1 + (- \lambda) \cdot 1 + 2 \cdot (-\lambda) + (-\lambda) \cdot (-\lambda) - 72 = 0 \newline 2 - \lambda - 2\lambda + \lambda^2 - 72 = 0 \newline \lambda^2 - 3\lambda - 70 = 0 \newline (\lambda + 3)(\lambda - 10) = 0 \newline \lambda_1 = -7, \ \ \lambda_2 = 10. \end{array} $$
First eigenvalue $$\tag{6} \begin{array}{rcl} 2 v_{1,1} + 9 v_{1,2} = -7 v_{1,1} & \Rightarrow & 9v_{1,1} = - 9v_{1,2} \newline & \Rightarrow & v_{1,1} = -v_{1,2} \newline & \Rightarrow & \displaystyle V_1 = \frac{1}{\sqrt{2}} \left[ \begin{matrix} 1 \newline -1 \end{matrix} \right] = \left[ \begin{matrix} 0.70710678118 \newline -0.70710678118 \end{matrix} \right] \end{array} $$ $$\tag{7} \begin{array}{rcl} 8 v_{2,1} + v_{2,2} = -7 v_{2,2} & \Rightarrow & 8 v_{2,1} = -8 v_{2,2} \newline & \Rightarrow & v_{2,1} = - v_{2,2} \newline & \Rightarrow & \displaystyle V_1 = \frac{1}{\sqrt{2}} \left[ \begin{matrix} 1 \newline -1 \end{matrix} \right] = \left[ \begin{matrix} 0.70710678118 \newline -0.70710678118 \end{matrix} \right] \end{array} $$
Second eigenvalue $$\tag{8} \begin{array}{rcl} 2 v_{1,1} + 9 v_{1,2} = 10 v_{1,1} & \Rightarrow & 8v_{1,1} = 9v_{1,2} \newline & \Rightarrow & \displaystyle V_2 = \frac{1}{\sqrt{145}} \left[ \begin{matrix} 8 \newline 9 \end{matrix} \right] = \left[ \begin{matrix} 0.66436383883 \newline 0.74740931868 \end{matrix} \right] \end{array} $$ $$\tag{9} \begin{array}{rcl} 8 v_{2,1} + v_{2,2} = 10 v_{2,2} & \Rightarrow & 8 v_{2,1} = 9 v_{2,2} \newline & \Rightarrow & \displaystyle V_2 = \frac{1}{\sqrt{145}} \left[ \begin{matrix} 8 \newline 9 \end{matrix} \right] = \left[ \begin{matrix} 0.66436383883 \newline 0.74740931868 \end{matrix} \right] \end{array} $$
comparion
| Approach | Numerical | Theoretical |
|---|---|---|
| $V_1$ | [0.74740932, 0.66436384] | [0.70710678118, -0.70710678118] |
| $V_2$ | [-0.70710678, 0.70710678] | [0.66436383883, 0.74740931868] |
| |
dot product
the_1 @ the_2 = [[0.05872202]]
num_1 @ num_2 = [[0.05872202]]
import numpy as np
the_1 = np.array([
[0.74740932, 0.66436384]
])
the_2 = np.array([
[0.70710678118],
[-0.70710678118]
])
the_dot = the_1 @ the_2
print("the_1 @ the_2 =", the_dot)
num_1 = np.array([
[-0.70710678, 0.70710678]
])
num_2 = np.array([
[0.66436383883],
[0.74740931868]
])
num_dot = num_1 @ num_2
print("num_1 @ num_2 =", num_dot)