What is a determinant?

How do we interpret the determinant intuitively? Well, here is one way!

This video was requested by Thecalculatorman on reddit!

A few quick notes:
* There are limitations to this way of thinking about the determinant, but for the most part it's solid for 3 and 2D objects.
* Finding the area of the transformed unit cube is the same as finding the area of the parallelpiped, just a little easier to explain. In hindsight, I should have added this definition too.
* There is a lot I skipped over, like how to perform the determinant. That wasn't the point of this video. I wanted to give people an intuitive feel for what the determinant was doing underneath.

As always, the simulations were done live on:

Feel free to follow me on Twitter!

And the music is from Josh Woodward (sped up 1.5 times):

Thanks for watching!

Also, discord:

[billsmyth]

It helps that the matrix is symmetric so that the eigenvalues are real and the eigenvectors are orthogonal. Not to knock, though. This is a beautiful demonstration. Generations of teachers have taught the determinant like it's just an arbitrary combination of numbers that somebody pulled out of thin air (to put it politely). The interpretation as a volume expansion is intuitive, and it also explains all those other interesting properties that the determinant has. For example, the det(A*B)=det(A)*det(B) - of course!. How about inverses? The inverse just gives you back the original unit cube, so det(inv(A))=1/det(A). And if A is singular? det(A)=0, so the cube gets squashed flat. So of course the singular matrix has no inverse, meaning that the squashed cube can't be reconstructed. Very cool :)

[techtana]

Your 3 minutes video just changed how I view matrix.

[blackheart]

I always used to try to understand what I was doing during calculating the determinant in the class. Now I could understand what I was calculating. Thank you so much! I wish may I had the teacher like you who could make me feel these concepts in bones.

[edoardosaccani]

The point is just that you are taking a linear transformation of rank n, from a vector space of size n to itself, such that all the eigenvalues are real (and all eigenvectors have period 1) which means that the matrix representing the endomorphism is diagonalizable over R. Then the important property is that the determinant is an invariant and so it's the same considering the matrix of the endomorphism expressed with respect to the canonical base and with respect to one of the bases which "diagonalizes the matrix". Then you can finish knowing that the determinant of a diagonal matrix is the product of the elements on the diagonal (aka the eigenvalues).

Just wanted to give an explanation on why it works, the video was great

[chil]

A lot of people don’t understand Mathematics because of lack of explanation like this!

[souravmukherjee]

I learnt much more in these three minutes than the entire semester class of linear algebra. It was really awesome and it gave me the feeling that I can see things instead of just solving mechanically

[kingvdarkness]

please get rid of the background music

[yongyoon]

This is Eureka moment. Determinant, Eigenvector, and Eigenvalue.

It's like after enjoying years of ham, bacon, and pork chops without knowing their relationship, one suddenly realizes they are all from parts of same animal. And this animal could give love and joy to the human as pet, and even a new life as heart valve.

Great inspiration. Thanks.

[vrushabhsingh]

After learning and using determinants, eigen values and eigen vectors for 5 years, finally understood what they mean!. this was some kind of enlightening moment for me, feels like now i have seen everything and know everything that i need to know lol. Thank you!!!

[LeiosLabs]

Hey guys, this video is meant to give an intuitive definition of the determinant. There are oodles of way to calculate it and I kinda assume that people watching this video have done a determinant calculation before. There are a few notes in the description, but I needed this video for certain videos in the future, so it was definitely worth doing.

How did you guys feel about the "More info" tags that popped up? Were they too much? I think it's a good way to cite previous videos, but if you guys have a better way to do it, let me know!

Thanks for watching!

[shama_k]

Amazing video!! I never ever imagined determinants and eigenvectors this way... Thank you so much 👌👌

[gnramires]

A note that might add to this video: aligning a volume V cube along eigenvectors, you get a scaled cuboid of volume V*det(M). Do arbitrary weird objects also scale in volume as det(M)? Yes: divide your object of volume V' into a large number of little cubes oriented along eigenvectors. After you perform your transformation, the little cubes will still be non-overlapping (assuming our matrix is full rank, that is, one-to-one), so you can just add their values to approximate the volume of the weird shape. As we increase the number of cubes, the volume before transformation goes to V', and after V'*det(M), just as expected.

[inothernews]

this is beautiful! I've taken linear algebra courses in college but there's so much meaning and intuition behind it that I've yet to discover!

[jh_esports]

This is genuinely mind-blowing. I never truly understood what a determinant actually IS, I just took for granted that it somehow exists. Eye-Opening video. Thank you!

[TheCoolcat]

This one video was enough for me to subscribe (after glancing at the other videos you have). Thanks a bunch!

[que_]

Very well explained, and kudos for the visualization of the concept!

[jannickharambe]

came here to understand determinants, now I also understand eigenvectors and values even more. Wow thanks

[ahmedelsabagh]

This is the first time for me to be able to clearly and visually understand the relationship between determinants and eigenvalues

[netllcn]

A perfect toturial, a terrible background music. Instructer, a good lecture does not need music, because mathematics itself is a beauty.

[artisticgamer]

This is good that you give a clear concept with a reasonable reality based example... I really enjoying you:)