I remember being interested reading physics and chemistry in school. Discussing questions like “Why are stars spherical in shape ?” with my friend and reasoning that gravitational pull will eventually make any other shape into a sphere.
We used to pester our teachers with doubts about anything and everything and they would patiently explain the concepts and encourage us to explore more.
We had discussed building a thorium enrichment plant (as it was supposedly available in beach sand in Alleppey — my home town) or a magnetic levitation train for school project but finally ended up making an alarm that rings when you cross a light beam. I have to thank my friend’s very practical tinkering skills that made it a reality.
Every concept was reasoned from first principles and we would try figuring out properties of substances based on their molecular structure all the way till organic chemistry made things a bit complicated.
Learning was slow and fun. you had time to chew on concepts for weeks.
This went on for a while but later there was a slow transition in academics. The difficulty of concepts started increasing, pressure to perform in examinations became real and eventually it became a rat race to get into a good university and from thereon get into a good job and move up in life.
Learning became fast, time-boxed and with performance expectations.
I do not exactly remember when I learned about Eigenvectors and Eigenvalues but I do remember being taught the formula and the process to calculate them. There was no chewing on the idea, no understanding of the concepts but just enough to make sure that you can learn the process to do problems of a pre-defined format.
But looking back, I wish I had taken a pause then. Not care about what happens in the rat-race. Just take two bites of eigenvectors a day and chew them on for couple of hours till they became soft.
I just did that a couple of weeks back. I cooked up a python script which simulates an application of eigenvectors.
- Created a set of points in 3-dimensional space which represent a semi-circle
- Found the covariance matrix for these points
- Calculated the eigenvectors of the covariance matrix
- Plotted the vectors with the points
Then I rotated these points by 30, 45, 90 degrees with respect to x axis and plotted the vectors.
As you can see, the eigenvectors of covariance matrix represents the directions in which the data varies the most. In fact the eigenvector with the largest eigenvalue is the direction in which data varies the most and so on.
Well even now I can’t claim that I understand eigenvectors completely.
But now I know them a little better. I approached them with a tool that I knew and with lot of time and could visualise them.
What I want to convey is this:
If you are a student, have some courage and patience to slowly learn ideas till you are satisfied.
If you are a teacher, a parent or someone related to education, show these students that it is okay to take time to learn a concept and that while they rush through academics they should take as many pauses as they want to appreciate the wonders of science and maths.
And even if you are not related to formal education, you can still take a pause and learn.
Because learning never ends!