Eigen Values & Eigen Vectors¶
- Eigen is a German word meaning “characteristic”
- Eigen vector is also called as Characteristic vector
- Eigen vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it.
- The corresponding eigenvalue is the factor by which the eigenvector is scaled.
- Consider a small sqare centered at origin
- It has number of vectors
- Now let us apply the transformation of vertical scaling and see the distortion in the shape.
- We can observe that few vectors undergo a transformation and few do not.
- Same way if we apply horizontal transformation, then few vectors will not undergo any transformation
- Now look the following shear transformation
- You can observe that the vector colored in blue is not undergoing any transformation
- Let's now focus just on three vectors and see the how these vectors transform
- Apply vertical scaling (By a factor of 2) and see the transformation
- Only Green color vector does not change either in direction or magnitude
- Pink colored one does not change it\'s direction (i.e it is in the same span), but the size is doubled
- These hrizontal and vertical vectors which do not change their span are the characteristic of this transoformation
- These charcteristic vectors are called as Eigen vectors.
- The value by which a vector has scaled is called as its eigen value
- In this example Eigen value of Horizontal vector is 1
- Eigen value of vertical vector is 2
- Let us now apply pure shear transformation on the sqare and see the tranformation
- Which is the Eigen vector here ?
- Horizontal vector.
- Apply pure rotation now
- As you can see this transformation does not have any Eigen vectors
In [1]:
import numpy as np
import numpy.linalg as lg
In [2]:
# Find the eigen vectors and eigen values of the matrix [[3, -2], [1, 0]]
X = np.array([[3, -2], [1, 0]])
val = lg.eigvals(X)
print(val)
val, vect = lg.eig(X)
print("Eigen Values are", val)
print("vectors are", vect)
[2. 1.] Eigen Values are [2. 1.] vectors are [[0.89442719 0.70710678] [0.4472136 0.70710678]]