Pure math of matrix makes people feel bored, and quickly drop out, however for those who figured out using these mathematics to solve real life problems, it’s a great fun to learn and keep learning.

From this 3B1B video two real life problems are demoed: one is Romeo and Juliet love equation and the other is the Schrodinger wave equation.

The love case is simple, x’s love is proportional to y’s love positively, while y’s is reversely proportional:

Applying a traditional mindset to try to solve x and y equations respectively is not easy. One can do lot’s of guess/trial and a savvy mathematician may jump to cos sin equations correctly, but it is not generic and just too stretching. Matrix provides a holistic perspective in that we can picture at certain moment the status of x and y can be described using vector, so in the following, we can draw the original positional vector, dissect to it’s x and y axis, we get the derivative vector (orange vector line):

Using matrix perfectly expressed this relationship beautifully and succinctly.

as to why the above equation holds, reference the below screenshot to solve it from the very beginning e polynomial theorem

Now comes the critical step:

It requires a little geometry and analytical equation bridging savviness to deduce from e^(Mt) to the cos sin matrix

So the way to visually view the dynamic love system between Romeo and Juliet is

Next, try my best on Schrodinger Equation, which actually is exactly same as the love system