# Prove that similar matrices have the same characteristic equations

• Apr 3rd 2010, 06:21 PM
nathron
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 $\displaystyle det(A-LI) = det(M^-1AM-LI)$. (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 $\displaystyle det(MA^-1M^-1-LI)$, 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! (Happy)
• Apr 3rd 2010, 06:31 PM
chiph588@
$\displaystyle x^{-1}$: $$x^{-1}$$

$\displaystyle \lambda$: $$\lambda$$
• Apr 3rd 2010, 06:51 PM
tonio
Quote:

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 $\displaystyle det(A-LI) = det(M^-1AM-LI)$. (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 $\displaystyle det(MA^-1M^-1-LI)$, 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! (Happy)

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

Tonio

Ps. Of course, justify all the steps in the above proof.
• Apr 4th 2010, 01:15 PM
nathron
Hey tonio,

Thanks for the reply. When you do the step $\displaystyle \det(M^{-1}BM-{\lambda}I) = \det\left(M^{-1}(B-{\lambda}I)M\right)$ , are you assuming that $\displaystyle M^{-1}IM = I$? (Doh)
• Apr 4th 2010, 04:44 PM
tonio
Quote:

Originally Posted by nathron
Hey tonio,

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

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

Tonio
• Apr 4th 2010, 04:53 PM
nathron
Thanks!

You've been very helpful. (Clapping)