# Polynomial of Jordan Decomposition

## Homework Statement

Let A be a real or complex nxn matrix with Jordan decomposition A = $$X \Lambda X^{-1}$$ where $$\Lambda$$ is a diagonal matrix with diagonal elements $$\lambda_1,...,$$ $$\lambda_n$$. Show that for any polynomial p(x):
p(A)=$$Xp(\Lambda)X^{-1}$$

$$p(\Lambda)$$ should really be the matrix with p($$\lambda_j$$) on its diagonal for j=1,...,n but I couldn't figure out how to make that matrix in latex.

## The Attempt at a Solution

I'm guessing there should be a way to take p of both sides and somehow extract the X and X inverse, but I can't seem to figure it out. Does anyone see anything? Thank you.

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tiny-tim
Homework Helper
Hi azdang! (for matrices etc in LaTeX, see http://www.physics.udel.edu/~dubois/lshort2e/node56.html#SECTION00850000000000000000 [Broken])

Hint: if the polynomial is A2, then A2= XΛX-1XΛX-1 = XΛ2X-1 = … ? Last edited by a moderator:
Hey tiny-tim. Thank you for your response. Quick question: Is it okay to assume p(A) = $$A^2$$? (This would be the polynomial p(x)=$$x^2$$ right?) Can we say that because the problem say 'for any polynomial'? Or is there a more general way to prove this?

Also, I do see then that $$\Lambda^2=p(\Lambda)$$, so that is all clear. Thank you very much. I'm just wondering if we should find a way to represent p(x) more generally?

tiny-tim
Homework Helper
Is it okay to assume p(A) = $$A^2$$?

I'm just wondering if we should find a way to represent p(x) more generally?

Yes, we need to deal with a general p(x).

So you do still need to prove it for p(A) = An, for any n …

my n = 2 was just an example for you. Alright, cool.

So, we could say p(x) = xn.

Then, p(A)=An=$$(X \Lambda X^{-1})^n$$=$$X \Lambda X^{-1}X \Lambda X^{-1}...X \Lambda X^{-1}$$.

All of the X's and X-1 will cancel out except for the ones on the edges so we are left with: p(A)=$$X \Lambda^n X^{-1}$$.

So, we know p($$\Lambda$$)=$$\Lambda^n$$. Can we say that that is equivalent to the matrix with p($$\lambda_j$$) on its diagonal where j=1,...,n because it is a diagonal matrix?

tiny-tim
Homework Helper
just got up :zzz: …

… So, we know p($$\Lambda$$)=$$\Lambda^n$$. Can we say that that is equivalent to the matrix with p($$\lambda_j$$) on its diagonal where j=1,...,n because it is a diagonal matrix?

Yes, of course (for Λm). (I don't think the examiner would even expect a reason for that )

And then prove that it's "additive", in the sense that if B = XΛBX-1 and C = xΛCX-1 then ΛB+C = … ? I'm kind of confused by this last part. I don't really understand the subscripts on Lambda or where B and C are coming from. Thank you again, tiny-tim, for helping me get through this :)

tiny-tim
Homework Helper
I'm kind of confused by this last part. I don't really understand the subscripts on Lambda or where B and C are coming from. Thank you again, tiny-tim, for helping me get through this :)

I mean, if B and C are real or complex nxn matrices with Jordan decomposition B = XΛBX-1 and C = XΛCX-1 and B + C = XΛX-1 where Λ ΛB and ΛC are diagonal matrices, what is the equation relating Λ ΛB and ΛC ? But we cannot be guaranteed that the X's in the Jordan decompositions for B and C are the same, can we?

tiny-tim
Homework Helper
But we cannot be guaranteed that the X's in the Jordan decompositions for B and C are the same, can we?

?? Assume they are … then prove the proposition … and then use it to answer the original question.

Well then, does $$X\Lambda_B X^{-1} + X\Lambda_C X^{-1}=X(\Lambda_B + \Lambda_C)x^{-1}$$?

Then $$p(\Lambda_B + \Lambda_C)=p(\Lambda_B)+p(\Lambda_C)$$, so the matrix would be the diagonal matrix with $$\lambda_B_j + \lambda_C_j$$ on the diagonal, j=1,...n, right? I'm not sure if I'm understanding all this, or why we are showing it is additive. Sorry!!

tiny-tim
Homework Helper
Well then, does $$X\Lambda_B X^{-1} + X\Lambda_C X^{-1}=X(\Lambda_B + \Lambda_C)x^{-1}$$?

That's right! Then $$p(\Lambda_B + \Lambda_C)=p(\Lambda_B)+p(\Lambda_C)$$, so the matrix would be the diagonal matrix with $$\lambda_B_j + \lambda_C_j$$ on the diagonal, j=1,...n, right? I'm not sure if I'm understanding all this, or why we are showing it is additive.

Because it's easy to prove the question for p(A) = Am, but not for p(A) = Am + Al

so we just prove it for Am, show that it's the same X for any m, and then use the theorem above to prove that it's true for any sum of powers of A, ie for any polynomial. Okay, I think I'm understanding, though I might have a hard time explaining it. One thing in what you just said that confused me, what do you mean 'show that it's the same X for any m'. I really can't thank you enough for walking me through this btw.

tiny-tim
Homework Helper
One thing in what you just said that confused me, what do you mean 'show that it's the same X for any m'.

I mean, for example, A2 = XΛ2X-1 and A3 = XΛ3X-1 (and it's the same X, which was worrying you earlier )

Oooh, okay. Yes, I can see right away that they are the same X!

So, I should really just show this for:
p(x)=$$\alpha_0+\alpha_1x+\alpha_2x^2+...+\alpha_nx^n$$ for scalars alpha.

P.S. I totally just did this ^ and I TOTALLY get it. Again, thank you very very much!!