Proof: Orthogonal Matrices Satisfy A^TA=I
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- Опубликовано: 18 сен 2024
- One way to characterize orthogonal matrices is to say that a matrix orthogonal if and only if A transpose times A is the identity matrix. In this video, we prove this result using basic matrix calculations and the definition of orthonormal vectors.
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My professor never explained these concepts. I blame the curriculum, but it's hindsight. There was too much time on proofs, and not enough time on problems. Recently, I discovered that it helps to put oneself into the subject, something our distinguished young man does well.
I love this video ! I think you could also also add that A has always a determinant equals to 1 or -1.
It would be interesting because there is the group SO_n(R) (orthogonal and det = 1). With a little work, you can find SO_2(R) and discover the rotation matrix !
AA^T = I is also true since the the rows of A also form an orthonormal set. The whole thing is a pretty brilliant result. Not have to compute the inverse of A using row reduction is a huge win computationally as it's much faster to transpose A than perform all of that Gauss-Jordan bollocks.
New to the channel and I love the content. Keep it up with the amazing uploads .
This is a great explanation of the algorithm of orthogonality. It helped me a lot in trying to understand the concept and its use. Thank you.
First time I read this theorem it didn't make any sense. Turns out it's really intuitive, thanks for making this video ♡
Your explanations are very clear!
Thank you. Great job! I just subscribed.
Awesome dude you cleared everything up for me! Thank you
New subscriber. Great content and explanations!
I wonder if you are a math major. I am just curious because I am a math major myself, and I find the few videos I’ve watched on your channel to be well-constructed and straightforward. Congrats on your channel!
Luis F
Yes, I'm a math major!
@@MuPrimeMath Awesome. I kind of figured that from how to present the topics. If you do not mind me asking, where did you go (or are going) to school?
I'm currently an undergraduate at Caltech
@@MuPrimeMath I have never looked at the math curriculum from your school, but from your videos, it is obvious that it reaches great depth. You are very knowledgeable, indeed; kudos to you! I will be graduating from a small university in NJ (FDU) this May, but I am sure I only got to scratch the surface of many interesting topics in mathematics. Anyway, I wish you good luck in all your future endeavors!
I like to connect with mathematicians from across the globe. Is there any way to connect with you (like LinkedIn) for future reference?
P.S. Sorry for the long comments 😅
@@luisfortuna7893 Yes, I found him on LinkedIn and sent him birthday greetings and praise for how he has clarified much of the math I took at Penn State. I don't think he wants me to reveal his name here, so I've now called him "our distinguished young man".
instant subscribe the best lesson on this topic thank you sir
This explanation is even better than the ones from Chat-GPT.
Spring Break already? Hey, does Fleming Hovse still have that stupid bell?
One of my favorite matrix identities is det(exp A) = exp(tr A). Any chance you'll cover that in an upcoming video?
Thankyou for such a good video
Of course , I'm here!
*Just Great* .
Thank you so much.
I have two requests:
At first, please be more active.
Second, please more Matrix (actually more anything, especially math for physics).
Again, thank you so much ❤️
What college and what background is this lecture from?
Thanks a lot, it really helped😃
Very nice ...❤ Sir
Here's a question: what would you say the determinant of a matrix represents? I know a few ways to use determinants, but I don't have a good sense of what it really means. I get the feeling that it's got something to do with how much volume (in however many dimensions) the vectors delineate, but that's more of a vague sense than an educated opinion.
I have a few videos about the determinant. Here is one: ruclips.net/video/A9eJdQt5quw/видео.html
That's pretty much exactly it. It's how much the matrix scales the unit n-volume. Of course it only tells you about the n-volume and not any lower dimensional volumes, so it's a little limited there. Something I can definitely say for certain is that not understanding this makes learning _anything_ about linear algebra absolute hell, which is exactly why the university course I took on LA didn't bother communicating it well. (Apologies for the minor rant. That was not a very fun class even if I've since warmed up to LA significantly since then.)
@@MuPrimeMath ... okay, there are two concepts in that video that have blown my mind: 1) that an ordinary nxn matrix describes a stretching of orthogonal unit vectors, and 2) that the determinant indicates the total amount of stretching. I'm going to have to process this for a while. Thanks!
@@angeldude101 I am frustrated by how poorly math is taught a lot of the time too. I haven't had to touch this stuff in decades, but I want to understand it even if belatedly.
"It's how much the matrix scales the unit n-volume." - Mind, re-blown. I'm still processing this.
Determinates were yet another concept I missed because I could not envision them. It's nearly a half century late, but I gradually realized that I must see the concepts. Thanks to our distinguished young man and others such as Grant Sanderson (3 Blue 1 Brown), I can see the concept. It even makes sense of nullspace, which was a mystery to me when I took linear algebra at Penn State.
Suppose I applied a coordinate transformation to the matrix that did not stretch the matrix, just rotate it. The magnitude of the determinant would not change but it could flip between positive and negative, right?
The sign of the determinant describes the orientation of the vectors relative to each other. Swapping two vectors changes the orientation, so if you switch two columns in a matrix, the determinant changes sign.
Rotation matrices always have positive determinant because rotations preserve orientation. As you said, the volume of the region spanned by the vectors does not change upon rotation, so the determinant of a rotation matrix is always +1.
Awesome thank you
Was this proof by inspection?
I'd say proof by definition.
Yes it is actually, and I made a video on it here: ruclips.net/video/dQw4w9WgXcQ/видео.html
For me the video is incomplete. You should also show that in case of square matrices the row vectors create orthonormal set.
That follows from the fact that if square matrices A,B satisfy AB=I, then also BA=I. Hence if A^TA=I, then (A^T)^TA^T = AA^T = I, meaning that A^T is also an orthogonal matrix.
Thank you