Example of Minimal Polynomial

Matrix Theory: We apply the minimal polynomial to matrix computations. For a given real 3x3 matrix A, we find the characteristic and minimal polynomials and evaluate p(A) and q(A) for p(x) = x^3 + x^2 + 1 and q(x) = x^2 + 1. Then we apply Bezout's Identity to find matrices X and Y such that Xp(A) + Yq(A) = I.

[philippelandry]

Yes sir! Thank you math doctor Bob, I wish my teacher was as intelligible as you are. An apple for you, cheers.

[MathDoctorBob]

Your welcome, and thanks for the kind words!

[VinayakGP]

Thanks again. Your videos are helping me a lot to get through my Applied linear algebra course.

[Wahrscheinlichkeit]

Thank you for this excellent presentation.

[MathDoctorBob]

Thanks! It depends on the undergraduate program. I'd have no problem putting this on a grad qualifying exam. - Bob

[MathDoctorBob]

You're welcome! It depends what you have. If you have the characteristic polynomial and its irreducible factors, the only question is the exponents. (Note: I'm no expert on the applied theory.)

A proxy for matrix multiplication is to pick a basis and see which polynomial in A sends each of the basis vectors to zero.

[ENTJ]

That neck size is something isn't it. were you ever a formula 1 driver?

Great teaching btw

[royteeyah]

Thank you sir for taking the time to put up this video. May you be blessed and may this video help those in need!

[simpsonsnerd]

Helped a lot with a problem I was working on. Thanks!

[rajatdash]

Literally in our exam they asked to find out minimal polynomial in a 1 mark multiple choice question. It is a kind long question

[wdlang]

i think your lecture is between undergraduate and graduate levels

very helpful to me.

[wdlang]

you are right! Jordan block! thanks a lot for pointing this out for me.

[peterernst]

good stuff! :DDD keep the videos coming bro

[VinayakGP]

Thanks for the great explanation Sir.
My question is, If the size of matrices is big, then it will become difficult to find the minimal polynomial by substituting the matrix and multiplying for various powers to see if we are going to get a Zero matrix. Is there any other way to find minimal polynomial?

Thank you.

[pauluk]

From the fact we are given matrix A, are we trying to use methods like this to get at a particular pair of polynomials m(x) and n(x)?

What would it mean if there is no distinct minimal polynomial for A?

In that case it looks like m[A] + n[A] = p[A]^-1

[rahulnaik]

Hello sir.. i would lik to see your videos on jordan canonical form defination as well as examples.. also exact definition of minimal polynomial

[vinvic]

Hello! Great video.
What are your thoughts on using det ( A - X Id ) to find the characteristic polynomial versus using det ( X Id - A) ?

[nikola]

Thanks you man, thank you so much i finally understood it! Much love <3

[makboocs]

How would you do this without taking determinants?

[wdlang]

if the characteristic polynomial of A is (x-1)^2(x-2), then i think the mini polynomial is (x-1)(x-2) definitely, why the possibility of (x-1)^2(x-2)