(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if two matrices are similar then they have the same eigenvalues with the same algebraic and geometric multiplicity.

2. Relevant equations

Matrices A,B are similar if A = C[tex]\breve{}[/tex]BC for some invertible C (and C inverse is denoted C[tex]\breve{}[/tex] because I tried for a long time to figure out how to get an inverse sign in latex but couldn't figure it out...).

3. The attempt at a solution

To show that two similar matrices have the same eigenvalues with the same geometric multiplicities, I need to show that their characteristic polynomials are the same.

Let A,B be similar matrices. Then,

A = C(inverse)BC

A-[tex]\lambda[/tex]I = C[tex]\breve{}[/tex]BC - [tex]\lambda[/tex]I

A-[tex]\lambda[/tex]I = C[tex]\breve{}[/tex]BC - [tex]\lambda[/tex]C[tex]\breve{}[/tex]C since C[tex]\breve{}[/tex]C=I

A-[tex]\lambda[/tex]I = C[tex]\breve{}[/tex][B-[tex]\lambda[/tex]]C

det(A-[tex]\lambda[/tex]I) = det(C[tex]\breve{}[/tex][B-[tex]\lambda[/tex]I]C)

det(A-[tex]\lambda[/tex]I) = det(C[tex]\breve{}[/tex]det(B-[tex]\lambda[/tex]I)det(C)

det(A-[tex]\lambda[/tex]I) = det(C[tex]\breve{}[/tex])det(C)det(B-[tex]\lambda[/tex]I)

Therefore, det(A-[tex]\lambda[/tex]I) = det(B-[tex]\lambda[/tex]I)

So, A, B have the same characteristic polynomials. This implies that they have the same eigenvalues with the same algebraic multiplicity. However, I do not think that this implies that they have the same geometric multiplicity because it doesn't same anything about the dimension of the eigenspace. Does it? Any suggestions on how I should try to show that the geometric multiplicity is the same?

So does anyone have any advice? I spent a while to get the first part, I just could use a nudge in the right direction for the next part... Thanks in advance.

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# Homework Help: Similar Matrices & Geometric Multiplicity

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