The immense utility of Eigen Values and Eigen Vectors

This post will describe the value of Eigen Values & Eigen Vectors to a Quantitative Researcher

Physical meaning:
Your data will be specified in a coordinate system; Original Basis [A]. You might seek to transform this data, i.e. apply a function to this data [Y], as part of your algorithm. Eigen Vectors and Eigen Values give you clues on how this proposed function will interact with data in your Original Basis. The transformation you propose will distort the original data in this coordinate space, i.e. change the direction of all vectors in this coordinate space (when real world data is plotted as a vector), with a single exception. The transformation will not change the direction of data points that lie along the Eigen Vectors of the transformation, these data points will merely be scaled. The extent of scaling is described by the Eigen Value; 1 - remain As-Is, 0.5 - shortens by 50%, 1.5 - lengthens by 50%, -1 - mirror image along the same line.

Practical use:
Your data will be specified in a coordinate system; Original Basis [A]. You might seek to transform this data, i.e. apply a function to this data [Y], as part of your algorithm. This transformation could be computationally expensive for large matrices. This computational expense can be reduced if we change the coordinate system to a New Basis [B]. This because of an exceptional property of Eigen Vectors. Eigen Vectors, of the function Y, are the optimal choice of this New Basis [B], because of the mathematical property that once you change the basis of your data from A to Eigen vectors of Y, then applying the function Y to your data is reduced to the same as multiplying your original data with a diagonal matrix, with each element in the diagonal matrix being the Eigen values of function Y. 

Compute by hand: 

Here is a link to a video that is an excellent explanation: https://www.youtube.com/watch?v=IdsV0RaC9jM

Compute using Python:
We can use numpy to determine Eigen Value and Eigen Vectors
#-----------
import numpy as np
eVal, eVec = np.linalg.eigh(matrix)

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