similar matrices question

[jazzman]

Hey everyone!
I need to prove or disprove the following:

If two matrices A and B are 10x10 over \(\displaystyle \mathbb{R}\), their rank is 2 and the numbers 3 and 4 appear in their eigenvalues, then A and B are similar.

Now, since I know they are 10x10 and their rank is 2 then each one has eigen value 0 with a geometric multiplicity of 8. Then we can get to a conclusion that each of the matrices have the same eigenvalues.

Eigenvalues: (a.m. = algebraic multiplicity ; g.m. = geometric multiplicity)

• 0 (a.m.=8, g.m.=8)[/*:m:55ltqq4f]
• 3 (a.m.=1, g.m.=1)[/*:m:55ltqq4f]
• 4 (a.m.=1, g.m.=1)[/*:m:55ltqq4f]

Also, their rank is the same, their determinant (0) and their trace (7).
So I cannot disprove them being similar but this does not prove they are similar either!

Please help!

 

[joeyjoejoe]

If they have the same eigenvalues and rank, wouldn't this mean the two matrices have the same characteristic equation? I think that is sufficient to say they are similar, I am quoting the wording of this proposition that I found:



Proposition 11.5.2

1. If A and B are similar, then |A|=|B|.

2. Similar matrices have the same characteristic polynomial, thus the same eigenvalues (with the same multiplicities).