Remember "The Matrix" movie ? "The Matrix" was a cyberadventure, full of kinetic excitement. I find our mathematical matrix also to be full of adventures and transformations. Believe me, it can take you to places. Mathematical matrix tackled me early, and it was just some rows and columns in the early days. But it kept on dazzling me in the later years, and one day I was reading a research paper on stability of microgrids where I had to go through some state equations to reach to the Lyapunov stability functions, I got myself into a solution of an ODE where e was raised to the power of matrix. There and then I knew I was into a problem. Actually, had I tested it with fundamentals, I would have found it as a beautiful thing rather than a problem. So here is to why matrix is a beautiful journey.


If you have a matrix, you can use it as a suitcase.

Yes, you can pack your things into a matrix. Before taking the matrix ride & going to places, lets get familiar with the matrix. Lets take an example here, once I had to rent an apartment for my job posting. There were many choices. Each apartment had its own feature. I had to find the one that fits my needs. Many choices meant lot of information to crunch. So, I thought maybe I could pack all the information into something.

Something that could hold all the information such that it would assist me to make a choice. Thats where I used the matrix as a suitcase to pack the data. Let me show it, how matrix packed the information.

Apartment Name Number of Rooms Kitchen Balcony Daylight Cost Security Market Nearby
Rupas 3 Yes No Good Heavy Poor No
Vyas 3 Yes Yes Best Moderate Good Yes
Prism 4 Yes No Bad Moderate Good Yes
Solti 3 Yes Yes Worse Heavy Tight No

Thats 5x8 matrix packing all the information I gathered for my apartment. See, matrix is just like a suitcase, it holds our information for us.

Taking a ride on a Matrix.

Me and my friend along with you, lets say we are on a location 20N 30E in a place. This is from where we take a ride on matrix. We all choose our own ride. I pick

$$Ride1 = \begin{bmatrix} -1 & 0 \\ 0 & 1 &\end{bmatrix}$$

My friend picked this matrix as his ride:

$$Ride2 = \begin{bmatrix} -1 & 0 \\ 0 & -1 &\end{bmatrix}$$

And let me guess did you take this one as your ride ?

$$Ride3= \begin{bmatrix} 1 & 1 \\ -1 & 1 &\end{bmatrix}$$

Our starting location is 20E 30N. Lets pack this information and we get:

$$Start = \begin{bmatrix} 20E &\\ 30N &\end{bmatrix}$$

My matrix ride took me to:

$$My destination = \begin{bmatrix} -20E &\\ 30N &\end{bmatrix}$$

My friend was taken by his ride to:

$$Friend's destination = \begin{bmatrix} -20E &\\ -30N &\end{bmatrix}$$

And let me guess you were taken to:

$$Your destination = \begin{bmatrix} 50E &\\ 10N &\end{bmatrix}$$

See matrix took us to different places. Actually, I had a 90 degree rotation from my location. And, my friend had a complete 180 degree rotation. And you had some random travel.

In Maths we perform matrix multiplication of our starting position with the ride matrix and it takes us to places. In fact our starting location is stored in a vector. 20E is the x-component of the vector while 30N is y-component.

On taking me to dest_x = -20E, dest_y = 30N, actually my ride transformed my x_init = 20E by vector

$$= \begin{bmatrix} -1 &\\ 0 &\end{bmatrix}$$

and y_init is transformed by another vector

$$= \begin{bmatrix} 0 &\\ 1 &\end{bmatrix}$$

Here is a little illustration by my sister; we 3 are on starting position 20E 30N & you can see where our ride took us to.

Taking a matrix ride. Image-credit: Alsu Adhikari

But people even do e raised to the power of Matrix, what actually is this ?

Do you remember what e is in maths, we almost always remember it as a constant of value 2.718....But acutally it is a function \(e^x\) and we write it in Taylor approximated series as: $$e^x = {1+ x + {(x^2)\over 2!}+{(x^3)\over 3!}+{(x^4)\over 4!}+{(x^5)\over 5!}+...}.$$ So, x is the input. And \(e^x\) gives us the output. We can put anything in the input and oneday somebody put matrix into it. Actually lets put any matrix into the sereis, you can visulaize there is lot of matrix addition and matrix to the power 0f 2,3,4 etc. It is just matrix multiplication multiple times and addition multiple times. It is trasforming itself or lets say it is going on an adventurous ride.

You can visualize it as lets say we had a naked Goku with us. Putting Goku into matrix is like, transforming Goku, lets say, he wore his pants now, then again putting him into matrix is like adding another transformation, this time maybe he got his shirt. Again putting him on matrix will transform him to having a belt and on on on. And we finally get a covered up Goku. Thats what e raised to the power of matrix does, it transforms the position, status, feature with things it acts upon.

Here is a simple illustration of it:

Transforming Goku Image-credit: Hakar