Oxford Linear Algebra: Eigenvalues and Eigenvectors Explained

University of Oxford mathematician Dr Tom Crawford explains how to calculate the eigenvalues and eigenvectors of a matrix, with 2 fully worked examples.

Watch other videos from the Oxford Linear Algebra series at the links below.

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.

You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.

[cll]

The only video on YouTube that explains both concepts in an intuitive way without compromising on the mathematical details

[juliussoldan]

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.

[manfredvonrichtofen]

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

[owen]

Seriously it's great having someone who can teach math

[Abhisar_Gupta]

What a fantastic explanation. Thanks a lot.

[jvdroid]

Man, that was the kind of video I was looking for, it was so good to watch. Thanks for you job!

[RCSmiths]

Always look forward to this quality information. Math with a fun and humble twist to it

[rogeriojunior]

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

[koioioioi]

Although im still only in secondary school watching this is very interesting, so i will continue :D

[pedrolironderobles]

As always, great explanations and very interesting content!!

[rameshpanta]

Great intuitive explanation at the beginning. I was waiting for intuitive examples on its applications. Thank you !

[nityambohare]

I am waiting since ages of linear algebra's good lecture and my patience pay off thank you sir

[owen]

Awesome Tom. I'll join Proprep as linear algebra I have this semester 👍👍

[pepemosquera]

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

[rasainsbury]

In the 3×3 example, the eigenvector associated with eigenvalue -1 should be (1, 3, -3)'

[ACC]

Would also be good to hear the geometrical notions of Eigen values and eigen vectors!!

[ihaveacreeplingdepression]

you are soooo tom rocks, got this chapter last semestre now we using those shits for rotation matrices, quadratic shit etc, and i realized that i got a lotta understanding problems on this topic, not i got the meaning of the det
in this case

[tamasburik]

Wish I could've had you as my linear algebra prof

[umehmoses]

Love you Doctor, wish to meet you someday

[mnqobimsizi]

I WILL BE USING THIS METHOD TILL I DIE, IT EASY THAN DOING THE GAUSSIAN ELIMINATION