# Thread: Prove that similar matrices have the same characteristic equations

1. ## Prove that similar matrices have the same characteristic equations

Hello,

I have been asked on an assignment to prove that similar matrices have the same characteristic equations. I believe that I would like to prove that $det(A-LI) = det(M^-1AM-LI)$. (L = lambda) The only idea I have had thus far is to take the inverse of the right hand side of that equation. The farthest I have reduced that to is $det(MA^-1M^-1-LI)$, but even if that IS right, I don't know where to go from there. Any help would be great.

Sorry about the exponentialized -1s. I don't know how to properly display an exponent in the math code.

Thanks!

2. $x^{-1}$: $$x^{-1}$$

$\lambda$: $$\lambda$$

3. Originally Posted by nathron
Hello,

I have been asked on an assignment to prove that similar matrices have the same characteristic equations. I believe that I would like to prove that $det(A-LI) = det(M^-1AM-LI)$. (L = lambda) The only idea I have had thus far is to take the inverse of the right hand side of that equation. The farthest I have reduced that to is $det(MA^-1M^-1-LI)$, but even if that IS right, I don't know where to go from there. Any help would be great.

Sorry about the exponentialized -1s. I don't know how to properly display an exponent in the math code.

Thanks!

$A\sim B\Longrightarrow A=M^{-1}BM\Longrightarrow p_A(t)=det(A-tI)=\det(M^{-1}BM-tI)$ $=\det\left(M^{-1}(B-tI)M\right)=\det(M^{-1})\det(M)\det(B-tI)=p_B(t)$

Tonio

Ps. Of course, justify all the steps in the above proof.

4. Hey tonio,

Thanks for the reply. When you do the step $\det(M^{-1}BM-{\lambda}I) = \det\left(M^{-1}(B-{\lambda}I)M\right)$ , are you assuming that $M^{-1}IM = I$?

5. Originally Posted by nathron
Hey tonio,

Thanks for the reply. When you do the step $\det(M^{-1}BM-{\lambda}I) = \det\left(M^{-1}(B-{\lambda}I)M\right)$ , are you assuming that $M^{-1}IM = I$?

Yes, of course...and also the fact that scalar matrices commute with any matrix, so $tI=tMM^{-1}= M(tI)M^{-1}$...and use also left and right distributivity of matrix multiplicatio.

Tonio

6. Thanks!