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()
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]
matrix
[ 2 9 8 1 ] [ v 1 v 2 ] = λ [ v 1 v 2 ] (1) \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]
[ 2 8 9 1 ] [ v 1 v 2 ] = λ [ v 1 v 2 ] ( 1 ) [ 2 9 8 1 ] [ v 1 v 2 ] = λ [ 1 0 0 1 ] [ v 1 v 2 ] (2) \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]
[ 2 8 9 1 ] [ v 1 v 2 ] = λ [ 1 0 0 1 ] [ v 1 v 2 ] ( 2 )
[ 2 9 8 1 ] [ v 1 v 2 ] − λ [ 1 0 0 1 ] [ v 1 v 2 ] = 0 (3) \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
[ 2 8 9 1 ] [ v 1 v 2 ] − λ [ 1 0 0 1 ] [ v 1 v 2 ] = 0 ( 3 )
[ 2 − λ 9 8 1 − λ ] [ v 1 v 2 ] = 0 (4) \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
[ 2 − λ 8 9 1 − λ ] [ v 1 v 2 ] = 0 ( 4 )
determinant ∣ M ∣ = 0 |M| = 0 ∣ M ∣ = 0 ( 2 − λ ) ( 1 − λ ) − 9 ⋅ 8 = 0 2 ⋅ 1 + ( − λ ) ⋅ 1 + 2 ⋅ ( − λ ) + ( − λ ) ⋅ ( − λ ) − 72 = 0 2 − λ − 2 λ + λ 2 − 72 = 0 λ 2 − 3 λ − 70 = 0 ( λ + 3 ) ( λ − 10 ) = 0 λ 1 = − 7 , λ 2 = 10. (5) \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}
( 2 − λ ) ( 1 − λ ) − 9 ⋅ 8 = 0 2 ⋅ 1 + ( − λ ) ⋅ 1 + 2 ⋅ ( − λ ) + ( − λ ) ⋅ ( − λ ) − 72 = 0 2 − λ − 2 λ + λ 2 − 72 = 0 λ 2 − 3 λ − 70 = 0 ( λ + 3 ) ( λ − 10 ) = 0 λ 1 = − 7 , λ 2 = 10. ( 5 )
First eigenvalue
2 v 1 , 1 + 9 v 1 , 2 = − 7 v 1 , 1 ⇒ 9 v 1 , 1 = − 9 v 1 , 2 ⇒ v 1 , 1 = − v 1 , 2 ⇒ V 1 = 1 2 [ 1 − 1 ] = [ 0.70710678118 − 0.70710678118 ] (6) \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}
2 v 1 , 1 + 9 v 1 , 2 = − 7 v 1 , 1 ⇒ ⇒ ⇒ 9 v 1 , 1 = − 9 v 1 , 2 v 1 , 1 = − v 1 , 2 V 1 = 2 1 [ 1 − 1 ] = [ 0.70710678118 − 0.70710678118 ] ( 6 ) 8 v 2 , 1 + v 2 , 2 = − 7 v 2 , 2 ⇒ 8 v 2 , 1 = − 8 v 2 , 2 ⇒ v 2 , 1 = − v 2 , 2 ⇒ V 1 = 1 2 [ 1 − 1 ] = [ 0.70710678118 − 0.70710678118 ] (7) \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}
8 v 2 , 1 + v 2 , 2 = − 7 v 2 , 2 ⇒ ⇒ ⇒ 8 v 2 , 1 = − 8 v 2 , 2 v 2 , 1 = − v 2 , 2 V 1 = 2 1 [ 1 − 1 ] = [ 0.70710678118 − 0.70710678118 ] ( 7 )
Second eigenvalue
2 v 1 , 1 + 9 v 1 , 2 = 10 v 1 , 1 ⇒ 8 v 1 , 1 = 9 v 1 , 2 ⇒ V 2 = 1 145 [ 8 9 ] = [ 0.66436383883 0.74740931868 ] (8) \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}
2 v 1 , 1 + 9 v 1 , 2 = 10 v 1 , 1 ⇒ ⇒ 8 v 1 , 1 = 9 v 1 , 2 V 2 = 145 1 [ 8 9 ] = [ 0.66436383883 0.74740931868 ] ( 8 ) 8 v 2 , 1 + v 2 , 2 = 10 v 2 , 2 ⇒ 8 v 2 , 1 = 9 v 2 , 2 ⇒ V 2 = 1 145 [ 8 9 ] = [ 0.66436383883 0.74740931868 ] (9) \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}
8 v 2 , 1 + v 2 , 2 = 10 v 2 , 2 ⇒ ⇒ 8 v 2 , 1 = 9 v 2 , 2 V 2 = 145 1 [ 8 9 ] = [ 0.66436383883 0.74740931868 ] ( 9 )
Approach Numerical Theoretical V 1 V_1 V 1 [0.74740932, 0.66436384][0.70710678118, -0.70710678118]V 2 V_2 V 2 [-0.70710678, 0.70710678][0.66436383883, 0.74740931868]
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)