Full Example: Diagonalizing a Matrix

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Let's compute a full example of Diagonalizing a matrix via eigenvectors and eigenvalues. The steps:
1) Compute the eigenvalues. Take the determinant of A-lambda I and set it equal to zero. The solutions are the eigenvalues
2) Compute the eigenvectors. For each eigenvalue solve (A-lambda I)x=0, which is a homogeneous system
3) Form the P and D matrix. The D matrix is just the eigenvalues down the main diagonal. The P matrix is the corresponding linearly independent eigenvectors for each eigenvalue as columns, in the same order as their eigenvalues were put into D.
4) Finally, we can test that A=PDP^-1 is true by multiplying out the right hand side.

Learning Objective: Starting with a matrix, compute the eigenvalues and eigenvectors and then finally diagonalize the matrix.

Course Playlists:

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This video is part of a Linear Algebra course created by Dr. Trefor Bazett

[TheRagingSun]

Hey Trefor. I've been watching your videos for quite some time now and I JUST realized that you're a professor at my University. Anyway great work. I hope be in your class for calc 3/4.

[pranavmahapatra]

3:20 Those of you confused about the free variables:

The first column has a leading 1 that corresponds to x1, so it's not a free variable.

Neither of second or the third column have a leading 1 "at the last row", that correspond to x2 and x3 respectively, so they are both free variables.

So, x2 = s, x3 = t

Now, first row: x1 + x2 + x3 = 0 + 0 + 0;
Second row: x1 + x3 = x1 + t = 0, we get x1 = -t;
Yeet the third row straight away.

x1 = -t, x2 = s, x3 = t

This general solution can be expressed in a vector form:

( x1 ) ( -t )
( x2 ) = ( s )
( x3 ) ( t )

You can decompose this as a linear combination like this (as shown in the video):

( x1 ) ( 0 ) ( -1 )
( x2 ) = ( s ) ( 1 ) + t ( 0 )
( x3 ) ( 0 ) ( 1 )

If you sum it up, you get the one I showed above.

[simenl]

Great video!
Btw just a tip for people:
When you have found P and D you can check if AP = PD instead of figuring out P inverse.
If AP = PD then you've found the correct values.

[tonystank]

Always here on the day before the exam! And I'm never disappointed!! Last semester; for Differential Equations... Now for Linear Algebra!! Thanks Doc!! Respectful!

[Brantendo]

I saw many videos. MANY. This one made SO MUCH SENSE to me. Thank you.

[georgesadler]

DR. Bazett thank you for the video on Diagonalization of a Matrix.

[therandomactivist]

Still confused about the free variables

[umgubularslashkilter]

Thank you for being so clear and engaging. You have interesting/challenging points, you don't shy away from repeating the things you have already covered again and you are making really clear what is the convention and what is something the textbooks do to save some space.
I think you just saved me hours of work searching for information!

[wattyven]

You were the best math lecturer I ever had at U of T - I'm still using your videos to help my brother through school now!

[habibnurmagovedov]

greeting from France you are helping students at the international level !

[RazorIance]

Hi, I'd like to ask for clarification regarding finding the eigenvectors, with respect to the values t and s.
In the second row of the matrix with the eigenvalues subbed in, which is 1, 0, 1, setting x3 = t means that x1 = -x3 = -t. But does that not imply that x2 = 0? So why would the column vector of s be (0, 1, 0)? I'm getting a bit confused there.

[therattleinthebook]

Sorry if this is obvious, but why did you do the P calculation at all? If you already knew that the diagonalized matrix had the diagonal equal to the eigenvalues, and you found the eigenvalues super early on, why did you need the eigenvectors?

[DiyaeDiyae-hicn]

Oh wow, first of all thank you for making this video because without you professor I would have put the pen down whenever I came across this question. Second of all, I don't study maths in English but in French and the video is just self-explanatory. Thank you for your time and have a nice day !

[sirjee]

Excellent teaching.... Thanks a lot Sir ji 👍

[peienshao]

Finally, a video neither skip x calculation nor keep mention how to calc with known P : ) Thank you sir

[Emilie_in_june]

Omg my final is tomorrow and this literally saved me

[kahtanalobaidi]

I am gonna have this problem on my final. this is a great help

[ricki]

I don't really understand in which order I put the eigenvalues in D. Is it always ascending order or what's the trick? Also does finding the eigenvectors have any relevance to finding the diagonal matrix D, since you only need the eigenvalues right?

[maureensikora]

Thank you! This is so much more clear than my professor.

[syremusic_]

I am so.... tired. Finals week. Thank you for your help Trefor🙏🏻