Powers of Matrices and Markov Matrices

MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
View the complete course: ocw.mit.edu/RES...
Instructor: Gilbert Strang
Diagonalizing a matrix also diagonalizes all its powers.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@georgesadler7830]

Dr. Strang really wants each student to understand linear algebra in all phases from top to bottom and from left to right. After the lectures are over he wants you to retain this information forever. These lectures really tell you the student the passion DR. Strang has for teaching.

[@AmandaSMaria]

the best teacher in the world!! Thank you professor Strang!!

[@galolgedi9314]

Amanda Maria

[@yujing7544]

“the best teacher in the world!! Thank you professor Strang!! ”， the same words from my heart.

[@ffvgdsg5584]

as a student of international university, which is not in US i've witnessed that most professors are using prof Strang's lectures as a pattern and the book particularly we use is intro to linear algebra G.Strang. That is another evidence of that he is currently the best professor in the world

[@Jeshtroy]

He teaches in a way that a 3rd grader could understand!

[@dengdengkenya]

Fantastic! I'm really enlightened by this lecture, though I can't say Professor Strang always gave such a clear explanation on each linear algebra subject.

[@paulchan6818]

A good lecturer that connects to his students.

[@jimnewton4534]

To compute the nth power of a square matrix A requires at most log(n) matrix multiplications, i.e. n^3 log n. (assuming cubic time matrix multiplication) Why? because if n is even I can simply square A^{n/2}, and if n is odd I can multiply A by A^{n-1}. If I have the Eigen decomposition then raising to the nth power is n^3 (just compute the powers along the diagonal of the diagonal matrix) and multiple on left and right by V and V^{-1}.

[@edwardhartz1029]

That was superb, I love how thorough you are. Now I understand why pii^(n)=a*(lambda1)^n+b*(lambda2)^n+... (For some a,b,c.. which can be determined using simultaneous equations.)

[@arvindvishwakarma4257]

Best teacher in the world

[@jonahansen]

What a great teacher!

[@jesusglez09]

i love this Thank you Prof. Thank you MIT

[@saeida.alghamdi1671]

Quite Interesting implication of the presentation!

[@devrimturker]

Excellent explanations

[@tgx3529]

Time 12:24, is it realy Markov matrix from Markov proces?? 0,8+0,31 and 0,2+0,71

[@checkout8352]

Thank you vey much

[@dengdengkenya]

What if all eigenvalues were less than one in the case of Markov Matrix example? Or is there any theorem that proves at least one lambda is greater than one here?

[@jordanhansen6649]

There is always an Eigen value that equals 1 in the case of a Markov Matrix

[@sauravnagar1745]

How does he write the values of eigenvectors so easily? I mean he doesn't even perform any mental calculations. Does anyone has any clue?

[@yohanshailu5620]

make it 1.75x

[@engineershmily]

please anyone can guide me how if U(k+1)=AU(K) then U(k)=A^k U(0) at time 3:25

[@jancirani2748]

Start with u(1). 1. U(1)= A.U(0). --> eq.1 2. U(2)= A.U(1) = A.A.U(0) (from eq.1) U(2)= A^2.U(0) Proceeding like this, It gives U(k) = A^k.U(0). Hope, it's clear

[@shrinivasiyengar5799]

@@jancirani2748 can this be said to be similar to how in the continuous time system X' = AX gives a solution X(t) = (e^At)X(0)

[@WolfixDwell]

@@shrinivasiyengar5799 Hi, prolly little bit late but: Solution to differential equation as your system dynamic is equal to x(t)= (e^At)x(0), so this solution coresponds to differential equation. And solving the e^At corresponds to e^\lambda_1 t + e^\lambda_2 t ... bcs of equality of characteristic polynomial equation

[@wuoshiwzm001]

professor Strang is getting old.... but always giant to me..

[@videofountain]

Everyone is getting older or the other alternative. Including you.

[@justinkim7743]

Thank you so much