Jordan Canonical Form

In this video I calculate the Jordan canonical form of a 3x3 matrix by presenting all the 3 cases that could occur. This approach is a bit nontraditional and is a bit longer than the usual approach, but it works most of the time is (in my opinion) easier to apply. A similar approach works for matrices of larger size. Enjoy!

Update: I got an e-mail about this, and it turns out that this approach only works if and only if you can solve the equations that I gave you, which should be most of the time anyway. An example where it doesn't work is for example the matrix A = [1 1/2 1/2; 0 1/2 -1/2; 0 1/2 3/2]. For that matrix, you'd have to revert to the classical approach of finding the Jordan Canonical Form.

[niazghumro]

30 minutes equal to 3 hours learning. Thnx dear for your clear depiction of idea.

[drpeyam]

Update: Someone sent me an e-mail about this, and my approach doesn't work all the time, but it still works most of the time. Basically, it only works if and only if you can solve the equations that I gave you. It doesn't work for example with the matrix [1 1/2 1/2; 0 1/2 -1/2; 0 1/2 3/2]. Most of the time you should be able to solve them, but if you can't, then you'd have to revert to the classical approach of finding the Jordan Canonical Form.

[zubin]

9:57 The use of "WTF" as "want to find" is ... very interesting :)

[viktorm]

I just want to add, that if you keep v1 in parameterisized form when solving (A-lamba*I)v2=v1, you can use this method to jordanize ALL matrices. This is because you can select the parameters such that the equation has a solution! The same process applies for determining v3 and so on.
Great video! :)

[aldurthedrowshade]

I wish more people were left handed because that means as you write across the board, you aren't blocking what you just wrote <3

[harwardw]

Nice!!!! Finally got an understanding of how to finalize the Jordan Block with repeated eigen values! thank you so much!

[ChristopherEvenstar]

Thank you so much! This is great help for my midterm. The theory flew over my head too.

[leonardromano]

Thank you! I always wanted to know how to compute this, unfortunately we didn't learn this in our Linear Algebra class (Instead we learned lots of other cool stuff like rings)

[li-pingho]

this short video teaches more than my professor's 3 hours lessons 🥰

[mercedesgomez]

Thanks you so much! I barely understand my teacher but i do love Algebra (a little bit nerdy but i do) you me make feel happy again. THANKSS

[mehmeteminm]

This video helped a lot... But after watching 5 from other channels with simpler explanation for the background :D

[devinlocke]

One day, someone will adequately explain to me how you decide where the 1's go.

[seoexperimentations]

Simply brilliant 30 minutes summing up 3 classes WOW, and btw do we always use the particular solutions for the vectors of the P matrix?!

[Jnglfvr]

The reason at 12:30 that you found 3 vectors that satisfy (A - I)*V2 = V1 is because (A - I) times the other 2 vectors are already zero.

[aneeshsrinivas]

Are there any matrices of the form [[(2a+b)/(2a+2b+2c), (b+2c)/(2a+2b+2c), 0]
, [a/(2a+2b+2c),(a+2b+c)/(2a+2b+2c),(c)/(2a+2b+2c)],
[0, (2a+b)/(2a+2b+2c), (b+2c)/(2a+2b+2c)]
] that do require jordan normal form to evaluate.

[tekaaable]

Thank you! You are such a good teacher!

[jimwyland]

omg, this guy is enthusiastic and charming

[tgotmbi]

Thanks much for the clear teaching !!!

[dimosthenisvallis]

I have question. In my class, we want to find an element of A^n for given A. I know that for diagonalisable A, we can right i.e. A^n(1, 1)= a(λ1)^n + β(λ2)^n + ... + s(λm)^n. where λ1, λ2, ..λm are the eigenvalues, and then try to determine the a, b, ..s. But what if i have an A of your 2nd or 3rd example? how can i find an element without calculating the whole A^n? Ive calculated the A^n=PJ'P^-1 from A = PJP^-1 by induction, but i dont know how to proceed now.

[stydras]

In class we oftentimes also used the notion of "algebraische und geometrische Vielfachheit" translating into algebraic and geometric multiplicity. Those oftentimes really shorten the calculations if you aren't interested in the transformation matrix. We also used Jordan-diagramms. Are they common where you teach?