What is a determinant?
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- Опубліковано 22 лип 2016
- 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:
/ leioslabs
/ @leioslabslive
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And the music is from Josh Woodward (sped up 1.5 times):
www.joshwoodward.com/
Thanks for watching!
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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 :)
Yeah. I opted to make the video short and focus on intuitive arguments. I should have left a little more room for discussion, but maybe that's what the comment section is for?
@@LeiosLabs ok great ... I had actually figured out previously that a determinant of a two by two matrix was a surface .... But tell me when you say that this division NEW VOLUME divided with OLD VOLUME, then my question is : Is the old volume "1"? Thanks if you have time to answer me.. (I am studying linear algebra by myself)
@@TheNetkrot yes it pretty much is.BASICALLY IT DEPENDS ON THE BASIS VECTORS.
Generally the standard basis vectors are unit vectors ( i cap,j cap,k cap).so the volume is 1
@@krishnasaikanigiri971 thanks for this
Yes.....
Your 3 minutes video just changed how I view matrix.
I'm glad it helped!
@LeiosOS same on my side. I just knew the déterminant of a 2*2 matrix would be the area of a parallelogram but I didn’t know it would be a ratio in higher dimensions. You have very interesting content
i have discovered we are living in a matrix, nothings real mate
Please tell me too
That's a pretty good movie.
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
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.
Yeah, that was the point of the video. I am glad it was helpful!
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.
Yeah! Honestly, I struggled with the same concepts until I looked into it. I'm glad it was helpful!
Now Jacobian is a piece of cake. For coordinate tranformations, like the transformation from polar coordinates (r, φ) to Cartesian coordinates (x, y), the transformation does not change the volume but the unit, to keep the volume same, one needs scale factor known as Jacobian. And it is no surprise to know Jacobian is just a determinant.
Good analogy sir, have a +1!
Ham and bacon come from the same animal? :o
Fled From Nowhere lol I didn’t know either
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
A lot of people don’t understand Mathematics because of lack of explanation like this!
teachers tell you to memorize the formula, legends explain the logic behind the formula
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!
Looks good! keep up the good work
youv'e done a good work,and i appreciate what understanding i took from you
thank you, i would be watching more of your videos later
more info is fine as long as it doesnt bother, so i approve :)
Amazing good ....very good...are you a mathematician ?
please make subtitles in Ukrainian
Thank you, this was amazing. You have taught me about something in minutes which I couldn't learn from hours of lectures.
what is the transform that you have applied to get the new volume?
The determinant matrix. I used it as a transformation matrix.
Thank you very much for your detailed explanation and the channel in general!
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!!!
This one video was enough for me to subscribe (after glancing at the other videos you have). Thanks a bunch!
I'm glad! I tried to make this one a way to understand the determinant using more physical arguments, which some people appreciated, while others did not.
The way I see it, there are enough purely algebraic explanations and proofs regarding the determinant. What is severely lacking are intuitive notions which help guide computation. I have heard of the connection between the change in volume and its effects on the determinant before, but these specific visuals(which must have taken a bit of work) helped cement the idea even further, especially looking at the transformation with regards to the eigenvector basis.
This is good that you give a clear concept with a reasonable reality based example... I really enjoying you:)
Amazing video!! I never ever imagined determinants and eigenvectors this way... Thank you so much 👌👌
I'm glad it was useful!
This has answered SO many questions, thank you!
Very well explained, and kudos for the visualization of the concept!
Thanks! I am glad you found it useful!
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.
This video is really great! Thank you :D
Finally I'm able to understand way more on what I'm working with on my linear algebra class. Thank you!
Simply exceptional! This is the video i wanted to see!!
Glad you liked it!
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!
After so many years, i finally understand. Thank you very much
THANK YOU SO MUCH LEIOS , IT MADE MAKING REVISION OF MATRICIES AND EIGENVALUES MUCH MORE INTUATIVE AND ENJOYABLE !!! :) :) :)
Yeah, it's super cool!
Thank you so much.your explanations are so beautiful.
I memorized the property that determinant is product of eigenvalues without knowing why, and this really explains it, Thank you!
Yeah! It's one of those things that's a little difficult to grasp intuitively!
very cool stuff, thanks for sharing!
Glad you liked it! =)
Great explanation! While I get the idea the determinate is the factor we scale the original one, but I'm still wondering how can a square matrix and its transpose have the same determinant intuitively? I can check the formula det(A) = det(A^T) by induction for a square matrix A, but how to understand the intuition behind it? Thank you!
so nice and elegant!
Thanks for the video. Is it so that when the determinant is negative, the volume of the object always reduces no matter what?
Also, if the determinant is nine , does that mean for any new volume to the transformation, the ratio of new volume to the old volume will be nine?
Love these short videos! Subscribed, what sort of videos do you have coming up? I'd love something with regards to Principal Component Analysis?
Yeah, PCA is on my radar. I'll bump it up the list, but no promises as to when it will be out (these videos take a while to make even though they are short).
Thank you so much for that I was strugling with it for a long time. Will you please make a video on Physical or Geomatrical meaning of trace of Matrix...
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!
I'm glad you liked it! A lot of time mathematical concepts are hidden behind some sort of cryptic formula or method when things could be explained much more intuitively.
Yes. That got me wondering. What about repeated eigenvalues, or singular matrices..? Intuition tells me that singular matrices will yield a line or point after the transformation, i.e. 0 volume. And does that also mean we are unable to get back our original cube since no inverse can be found? Hm I am not so sure about repeated eigenvalues because sometimes I could find enough eigenvectors but other times when I can't, I'll just add a 't' in front of it (when solving ODEs). And what does THAT mean geometrically? Interesting stuff! Could you shed some light or share some sources that would? Thanks!
This is the first time for me to be able to clearly and visually understand the relationship between determinants and eigenvalues
seriously such a beautiful video with good description
I'm glad you liked it!
yeah mind blowing videos u have,which made people like me curious
Thanks for the great explanation. 👍
Well, done. Swift, concise, yet clear. *thumbs up
I'm glad you liked it!
Thanks, made me link the determinant from the eigenvalue matrix with the determinant of the matrix !!
Few words , much more understanding .
just amazing !
I'm glad it was useful!
came here to understand determinants, now I also understand eigenvectors and values even more. Wow thanks
tutorial was very helpful, thank you :)
I'm glad you liked it! I tried to keep it short and explain the determinant intuitively instead of going through the math.
Excellent Video!
Thanks, I'm glad you liked it!
An amazing video. Thank you.
Glad you enjoyed it!
A perfect toturial, a terrible background music. Instructer, a good lecture does not need music, because mathematics itself is a beauty.
Hi! Could you tell me how are you importing LaTex to a video editor?
very good video!
I love ur videos
we learned linear algebra for 1 semester and now i finally know what all of these things mean in 3 min.
You helped me in getting sense of my high school matrix!
I just started leaning vectors and this makes me wanna scream but it does help me to understand in a way so thanks.
I'm glad it was somewhat useful. Sorry if it was a little complicated!
my friend im in serach of this Q -> Is time the determinant of all events in the enviroment?
but why do we compute the determinant of 3x3 matrix like that? is there any reason of hiding rows and columns and alternative + and -?
Excellent!
I'm glad it was helpful!
I find this algorithm for the computation very intuitive.
Ty
Ur vdeo speaks volume 😄 thanks alot
since -3 is isolated in its own row and column for the determinant you could have just taken the determinant of the matrix at the top left times its cofactor at the bottom right (-3), giving you -3(1*1-(2*2)) = 9
Originally I though you would use the cofactor trick since the matrix was so nicely set up for it but since you didn't I thought I'd mention it
Sweet and simple
The last determinant where he got a 9 right? It was all inside one matrix so what was the original dimension and what are the new dimensions of the cube?
i think its relative to the identity matrix
I don't understand what the initial matrix acts on? on a cubic equation? how is the equation of a cube expressed with a matrix?
does this generalize? is the determinant of a 2x2 increase in area and whatever it's called for 4 dimensional objects. What if the object you are transforming is not a cube? but some other arbitrary shape. Does it still work?
Do you think you can do something like that a about hessian determinant? I mean, the volume of how much a 3d function is curving up or down (at least this is how I see second derivatives).... why is it negative when its a saddle point and positive when its a extremum? and why do we check determinants of each submatrice in n dimension?
Hmm. I am not sure myself. I would need to look into it! Thanks for bringing it up!
So, if any vector is having some x,y,z eigenvalues then the determinant of that matrix will be xyz
Am I correct
Wow thank you so much
Your voice is similar to the welchlab tutor , he is absolutely amazing , Especially the way he opened the complex number world in my eye.
I love that guy! His videos are great!
Truly a genius !
Amazingg!!
I'm glad it was interesting!
I wish I had this video back in school days
brilliant sir
What happens when you apply a Matrix Transformation whose det=0 to a unit cube? What will be the resultant cube look like? Is it that there will be infinite possible resultant cubes with infinite shapes?
Hi Leios
I have basic question dont think its silly question. why we need matrix ? what is the application ?
in schooling we were thought have to perform matrix operations but not the application
Machine learning uses Linear Algebra (matrices and vectors) extensively!
and this is only to name one application which is already transforming the world!
Ur a very unique teacher
I, m a bit confused can you please tell me why did you or how did you multiply the matrix with the cube
This video describes ways in which we can understand the determinant. One way is the area of a transformed cube after application of a transformation matrix to each unit vector.
How would you interpret negative determinants then? In this particular example.
Basically the cube moving in the other direction, if that makes sense.
It means the cube was not only rotated and scaled, but also mirrored by the transformation. (The transformation transforms a right handed system of vectors to a left handed one and vice verca.) The change in volume is actually given by the absolute value of the determinant.
Think of the cube shown in the video is above the surface. A negative determinant would indicate a cube below the surface mirroring the one with the positive determinant.
What a POV changing video!!!"❤
This is so beautiful I wanna cry
I'm glad it was useful!
how could u align a cube in the direction of eigen vectors ? Are eigen vectors of any matrix are mutually orthogonal to each other ?
aravind gopal yes eigenvectors are basis spanning the eigenspace, they are linearly independent of one another thus orthogonal too.
@@budasfeet Eigen vectors are not orthogonal to each other unless the A matrix is symmetric, which is the case here in this example. Second, linear independence of of two vectors (a) doesn't depend on them being orthogonal, (b) and can still span the entire 2D space without being orthogonal.
Good job, thanx bro.
I'm glad it was useful!
Can't get it , what are you guys doing in the beginning , are you are multiplying matrix with cube?
Thanks 😊😘
I was never taught this when we learned about determinants. We were only taught how to find one, not what it actually was.
Exactly. That was why I made the video
brilliant
incredible, I was wondering for so long what was the meaning of a det. Ty
really a great video, just changed the point of viewing matrix.
Here we are told to mug up that product of eigen values is the determinant of a square matrix. Thanks for telling why as well.
In single Matrix A How can we find volume old and new?
So in other words the determinant of a Matrix is the volume of the transformed unit cube in that matrix space 👍 After many years I finally get it )) And now I get how it's useful, like normalizating a matrix vectors by dividing the elements by determinant? Like we do with vectors x,y,z/length
even the number of elements are 9 in that matrix xd...awesome vid i was curious about the reason behind determinants and what they are used for...this video made it so much easier to understand cuz my teacher just plays around with properties what i lacked is the reason to use them...but i am still curious those elements in the matrix what do they represent in terms of the cube?
hey thanks for this
Can anyone tell me. What is the acutal meaning of determinent?
if we take a number of vectors in a vector space, they (at least in 2D and 3D) form a shape. Well, we seek an answer to the question: "by how much the area is scaled"? Therefore, it makes sense to associate to every transformation a unique scaler value, denoted by det(T). Noting that Det: a number of vectors ---> scaler field.
very nice
I'm glad it was helpful!
Wasn't what I was looking for but mind blown anyways
I understand the criticism. This video is probably one of my more controversial ones because it is trying to give an intuitive description of the mathematics instead of showing the math, itself.
What I don't get ....whats the difference between a norm ..and an eigenvalue ...if they both scale and stretch
Is it useful only if it is Affine transformation??
I don't know much about affine transformations, actually.
does it mean that the determinant of a matrix (in dimension 3x3) tell us how much can we magnify another matrix (also 3x3 representing a cube) if we multiply the first one by the second??? If this is it, it´s astounding awesome!!!
Hii but how to multiply a cube by matrix
As someone who has suffered from matices for years,and I mean yeeeeeeeears,,Thank you
I'm not understanding anything.
I understand that this one is a little hard to follow and will avoid this format in the future. The idea of this video was to describe how to calculate the determinant in a new way for those who have been doing the calculation their whole lives.
Your video taught me more than my Linear Algebra class on this subject.
Marvellous
I wonder when Gauss had work in matrix. Did he have this geometric description in mind?
its very fun and easy to prove this with a 2x2 matrix and two vectors u and v that will undergo a transformation. Just calculate absolute value of det(u,v) to find the old area, then calculate the new area: absolute value of det(T(u),T(v)). Then you will easily see after some algebra steps that this new area is equal to absolute value of det(A)*old area
thanks a lot for info
Thanks a lot for watching!