Layman explanation for Eigenvectors and Eigenvalues.
Find the nearest pen/pencil. Roll the pen between your palms such that when it spins, the axis of rotation matches the same vector that the pen points. Now assume we have a 3D simulation that rotates the pen in this way. In the simulation of the rotated pen, the computer has to calculate the position of each point within the pen. The rotation is performed by a 3D transformation matrix that when multiplied by the matrix of the points in the pen, defines precisely how that pen will rotate on the 3d cartesian plane. The pencil is just a 3D matrix. There is another matrix that when multiplied, yields the correct rotation around the axis of rotation.
In this little pen rolling simulation, you have the Matrix for the location of particles in the pen, and you have the matrix that says exactly how to do the 3D transform to make it rotate. What if you wanted to know the axis of rotation of the pen, given only the pen and the transform. You would not know how to do it.
Enter stage left the following equation:
np.dot(MATRIX, vector) == multiple * vector, which would translate to this in our little story:
np.dot(PEN, TRANSFORM) == SOME_NUMBER * TRANSFORM
This equation asserts that if you can find a number times the transform that is the same as the dot product between the pen and the transform, that yields the axis of rotation. You can find out which way the pen is pointing given only how the particles in the pen spin. It's a clever trick to isolate variables and discover new truths.
OK but why
This methodology is useful because discovering how a Matrix times vector produces the same result as a scalar times a vector helps us find the axis of rotation that minimizes variance. Enter stage right principal component analysis algorithm. An algorithm that reduces dimensionality while minimizing the reduction of information.
The method you use to discover Eigenvectors are the methodology that Principal Component Analysis uses to find the new basis vector for the points scattered on the coordinate plane. Imagine we have two input features. Tire age and tire wear. We plot these on a scatterplot and it makes a diagonal straight line.
Eigenvectors and eigenvalues play a role when we want to compress tire age and tire wear since they are so plain. Imagine a 2d scatterplot with points on a straight diagonal line. This 2 dimensional straight line can be compressed into one dimension without much data loss. So find the eigenvector of the points, that is the axis of rotation, so imagine taking a pencil and rolling it between your palms, it spins along its axis of rotation. The eigenvector is that vector of axis of rotation of minimum variance. You can rebase the points around that vector, and you've compressed 2 dimensions to one dimension. We're happy because we've reduced data size but not decreased information gain/variance.
Reset the cartesian plane to put the x axis along the Eigenvector, and you have a recipe for compressing a 2 dimensional object to one dimensional object, while also preserving all of the information contained in the 2d object.
Now that you have this algorithm and procedure in mind, you can checkout 3Blue1Brown's explanation and hopefully the beauty of this tool will materialize. https://www.youtube.com/watch?v=PFDu9oVAE-g If it does click, you'll be able to explain how finding the Eigenvector is a great tool for helping address the curse of dimensionality as it applies to neural networks and other supervised learning algorithms.
Even simpler words, if you're still confused:
Surely you have thrown and caught a ball in midair before. Have you wondered how your brain is able to send instructions to the muscles in your shoulder, biceps and forearm to throw and catch a ball in midair even though you are unable to describe the calculus III of multiple variables, of what must be happening to perform these miracles? Eigenvectors and Eigenvalues are structures that your brain uses in order to correctly access the incoming trajectory of the ball, given only 2D frames over time. Your mind is able to untangle 2 dimensions into a 3 dimensions correctly. Your brain is about 2 billion years old and this functionality is present even in rodents and insects, so these principles are ancient. Your brain furiously processes the equation I described above. When a 3D array of particles, transformed by a matrix, equals a scalar times a numeric multiple, that satisfied equivalency is the target trajectory.
Over the 2.1 billion years or so, that nanotechnology in your head has stumbled across and harnessed the powerful concept of Eigenvectors and Eigenvalues. It's used all over, even during the process of learning and model compression during sleep. These Eigen-concepts are some of the core algorithms for why machine learning algorithms work. Tesla's self driving car team was able to harness the mathematics of these Eigen-transforms and apply them to the mathematics of optics and camera calibration, to transform 2D visual images into new ground truths for the position, distance, orientation, direction, velocity, and acceleration of vehicles and objects in the vicinity. These Eigen-structures are the tools of computer vision developers doing truth extraction from live photographs, which from the computers perspective is just a 2D matrix of numbers representing color.