How the Diagonalization Process Works

Description: Our goal is to fix a diagonal matrix that is similar to our given matrix. Why? Diagonal matrices are simple, and if two matrices are similar then they share many properties like the same eigenvalues. So similar to a diagonal matrix will allows us to quickly compute many properties. In this video we go over the process and explain why it works.

Learning Objectives:
1) Given a matrix, find an invertible matrix P and a diagonal matrix D so that A=PDP^-1; that is, A is a diagonalized.

Course Playlists:

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

[calculer]

This was exactly what I was looking for. Thank you for giving a clear, succinct explanation

[evgenyk.]

I am scrolling through some of these Linear Algebra videos to refresh my memory and to get some geometrical sense for eigenvalues and diagonalization (need to do some quaternions and Euler angles work). I find Dr. Bazett's explanations very intuitive, useful and well done. Thank you a lot.

[cutestbear]

going through Dr. Bazett's linear algebra course again and I think this is one of the most important lessons in the playlist

[xX_swagger_Xx]

This is amazing! Finally someone who actually explains why this is true instead of just stating the formula!

[redwoodenjoyer]

Thank you. Best video out here to finally help me understand this.

[anondoggo]

Taking a summer course and this is so useful for reviewing :). Thank you so much!

[nextzdota]

Idk if this counts as complete proof, but dang it filled a hole in my brain. Solving diagonalization problems is easy, but it's very nice to gain some intuition.

[kamalbhatt]

sir your videos are very helpful. I hope one day you get millions of views. Your way of teaching is the best for me. Once again thnx a lot

[Nucleardoom]

Great videos. Would you mind reordering the section on similarity to come before these as it seems you discuss similarity before diagonalization but this isn't apparent from the order the playlist is in. Thank you.

[robertusbellarvino]

can you please explain how that "A=PDP^-1" becoming "AP = PD", from my thought i just multiply P to both side, but to the right end, so P with P^-1 on the right side is cancel, but is that mathematically right ?

[utkarshgupta]

P is supposed to be n*n matrix..right? As P is initialised as invertible

[Rahul-uksu]

u didnt talk about similar matrices in linear algebra playlist

[EW-mbih]

Why is it AP=PD? I would say it’s AP=DP