Oxford Linear Algebra: Eigenvalues and Eigenvectors Explained
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- เผยแพร่เมื่อ 1 ต.ค. 2024
- University of Oxford mathematician Dr Tom Crawford explains how to calculate the eigenvalues and eigenvectors of a matrix, with 2 fully worked examples.
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Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: www.proprep.uk...
You can also find several video lectures from ProPrep explaining the topic further here: www.proprep.uk...
And fully worked video solutions from ProPrep instructors are here: www.proprep.uk...
Watch other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): • Oxford Linear Algebra:...
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: • Oxford Linear Algebra:...
What is the Determinant Function: • Oxford Linear Algebra:...
The Easiest Method to Calculate Determinants: • Oxford Linear Algebra:...
The video begins by introducing the eigenvalue equation which we are trying to solve, with a discussion of possible methods of solution. We see that the only way a non-zero eigenvector can be found is if the determinant of the characteristic matrix is zero, which gives us the characteristic equation, or characteristic polynomial. Solving this equal to zero gives the eigenvalues, which are then substituted back into the eigenvalue equation to give the corresponding eigenvectors.
The method is demonstrated first with a 2x2 matrix example, and then for a 3x3 matrix. In both cases we consider a general eigenvector before choosing one parameter to make the final vector as simple as possible.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac....
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Here we go eigen.
I laughed
Lol
lmao
The only video on TH-cam that explains both concepts in an intuitive way without compromising on the mathematical details
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Just wrote my Linear Algebra 2 exam yesterday at UWaterloo. Admittedly, I had more of a love and hate relationship with these 2 courses, but near the end and looking back at them, I did really enjoy them. Seeing these videos and actually being able to understand what's going on just makes me realize how far I've come, and if I could go back in time I would definitely take them again.
Amazing!! Congrats :)
Always look forward to this quality information. Math with a fun and humble twist to it
What a fantastic explanation. Thanks a lot.
Man, that was the kind of video I was looking for, it was so good to watch. Thanks for you job!
I am finally at a point at which I can use your videos as guide, not just as interesting videos about Math I dont understand :D
Your voice is so nice to listen to…
i wish i had learned it like that, but instead, the first time i saw it, was just proving hard theorems about existence of eigenvalues and eigenvectors, or orthonormal basis of eigenvectors or something like that, never had the time to actually play with the characteristic polynomial and find actual eigenvalues, great video
Seriously it's great having someone who can teach math
As always, great explanations and very interesting content!!
Although im still only in secondary school watching this is very interesting, so i will continue :D
In the 3×3 example, the eigenvector associated with eigenvalue -1 should be (1, 3, -3)'
ok there's one thing I'm slightly confused by. Where does the 6-landa come from? in my head that became 5 landa instead of 6 (I know this video came out one year ago but I'll shoot my shot.) I know this is a very simple thing to understand but I still don't really get it so yeah if anyone sees this and can explain it, it would be greatly appreciated 🙏
Great lecture! saw you at the duocon and decided to take a look to your channel, and this video was exactly what I needed for my algebra course, hoping to see diagonalization soon
Would also be good to hear the geometrical notions of Eigen values and eigen vectors!!
But 3blue1brown is so much better for that!
@@TomRocksMaths Aha!! Love the shoutout to Grant!! I was hoping to get your thoughts on geometrical notions of EV's!! But regardless in recent times, I definitely watch 3Blue1Brown along with your videos to get a complete overview of the topics!! Recently, I have started watching Dr.Steve Brunton's (Associate Professor, University of Washington) channel too!! Afterall, the more insights that I get... the more I appreciate the lore behind the math!!
Ok, the linear algebra is fine but the factorisation is basically witchcraft. I'll ask my son to teach me.
I WILL BE USING THIS METHOD TILL I DIE, IT EASY THAN DOING THE GAUSSIAN ELIMINATION
This video has been the closest I've ever come to understanding this thing
at 8:38, i thought it supposed to be landa-5 at the first entry?
Nice video, I love linear algebra!
Own values? 😅🤔🤷♂️
Wish I could've had you as my linear algebra prof
Does he talk about spectral values? SvD?
Awesome Tom. I'll join Proprep as linear algebra I have this semester 👍👍
Love you Doctor, wish to meet you someday
gracias.😉👍🏾
merci frere je va retour sur ce video a bintot!
Great lecture! 🙏🏻
you are soooo goooooodd
I was practicing elementary row ops for the 3x3 example, so I did r3 + r2/lambda. You end up getting an extraneous soln of 0. Btw how do you work backwards to get the A matrix from a given Eigenvalue and Eigenvector? Eg, lambda = 2, and (1, 0, 0) like the example. Or are these vectors sort of just for some characteristic of the matrix that is useful to us?
Brilliant. But couldn't we just argue that the determinant of a transformation matrix represents the scale factor applied to the modulus of a vector? So if we want a result of zero when applied to a non-zero vector, we need a determinant of zero?
10:28 Why is the general vector a 2×1 and can it be a 2×n?
It would be n x 1 in the general case of a n x m matrix.
Great intuitive explanation at the beginning. I was waiting for intuitive examples on its applications. Thank you !
I have a question. z in every case to be any value ?
🙌🙌🙌🙌
I am waiting since ages of linear algebra's good lecture and my patience pay off thank you sir
That 30 second explanation.
a cool maths teacher doesnt exis- 😳
Man, this is amazing.
You had written z=4z then went on to write z=0 . How ?
If you subtract z from both sides you get 3z=0 and so z=0
Would you state at the beginning that vector v is not zero? In an assessment situation?
An eigenvector is by definition nonzero.
@@rogerlie4176 thank you
9:10 there’s a shortcut here where you can just put, where lambda = m, m^2 - (sum diagonal)m + detA which instantly gives the characteristic equation. I’m this case sum diagonal = 6, detA = 8. 10:06 better to complete the square.
unfortunately, this simply goes through the simple mechanics of determining values - zero explanation, insight, into what these represent. Frankly, could have gotten a smart 14 year old to do this video. You need to up your game mate, stop appealing to middle of the road engineering students.
Sod off