Similarity Transformation and Diagonalization

In this video we investigate similarity transformations in the context of linear algebra. We show how the similarity transformation can be used to transform a square matrix into another square matrix that shares properties with the original matrix. In particular, the determinant, eigenvalues, trace, and rank of the two matrices are the same (and the eigenvectors of the similar matrix are related to the eigenvectors of the original matrix).

We then investigate a very specific similarity transformation that can be used to diagonalize the original matrix and place the eigenvalues along the diagonals.

Topics and timestamps:
0:00 – Introduction
0:55 – Definition of a Similarity Transformation
2:00 – Property 1: Same Determinant
5:17 – Property 2: Same Eigenvalues
10:50 – Property 3: Similar Eigenvectors
19:26 – Property 4: Same Trace
22:03 – Property 5: Same Rank
30:36 – Diagonalization
44:55 – Example 1: Non-Defective Matrix
54:27 – Example 2: Defective Matrix
58:16 - Conclusions

#LinearAlgebra #MatrixMath

[chayweaver.]

AE512: I like how these videos are bringing linear algebra back down to earth. This was the class in undergrad I struggled with the most because it's so hard to conceptualize. Starting to all make sense now, though

[zaneyosif]

AE512: Good to get a refresher on the properties of matrices and their transformations!

[princekeoki]

AA516: Great refresher on the properties of matrcies which have undergone similarity transformations!

[aimeepak]

AE512: This was a great refresher on similarity transformations :)

[princekeoki]

AA516: Good refresher on the shared properties of transformed matricies!

[yaffetbedru]

AA516: Thanks for the video. It was nice to see all the properties of similarity transformations

[Kumky]

AA516: Thank you for the review on matrix properties and similarity transformations!

[charlesharmon]

Great foundation for the topics that immediately followed.

[rowellcastro]

AA516: Great refresher. Thanks Professor!

[eswyatt]

Worked through the PDE playlist and made 33 pages of notes. Looking forward to this new playlist in linear algebra!

[WalkingDeaDJ]

Jason-AE512: Good Example and well structured lecture, it help me to understand the Similarity Transformation

[milesrobertroane]

AA526 Nice video! Always good to get a refresher on my linear algebra. Feels like I took Math 208 last century lol

[samialsharari]

Thank you for the best and most coherent explanations. I have a question, at 8:24, are we supposed to write the identity matrix again? (A- Lambda*I)

[NikolajKuntner]

I did my physics masters on quantum gravity and my PhD on plasma chemistry. Then I went to work for a silicon valley company. Now I just do QR decompositions and shuffle around C++ class instances all day...

[Gholdoian]

AE 512: For the diagonalization part, are we just transforming the A matrix so its mapped to a basis of its eigenvectors? So A then transforms into a matrix of its eigenvalues?

[daniellerogers]

AA516 I assume the benefit of arranging the eigenvalues across the diagonal lends to easier linear algebra operations later on?

[nithinadidela]

AA516: I should have definitely watched this before jumping to Controls26

[newguitarstudent]

Property 3: Similar Eigenvectors = Similar Right-Eigenvectors ? There is a proof for Similar Left-Eigenvectors?

[CAGEFIGHTER]

I gotta say; Your channel is very interesting! I also own a ford transit and I’m planning to convert it to Recreational van someday. So watching your van videos are giving me ideas. Plus the calculus lessons are superb!