# Homework Help: Finding X of (X^-1)AX = B

1. Jan 24, 2012

### smerhej

1. The problem statement, all variables and given/known data
Let A =

\begin{bmatrix}
\lambda & a \\
0 & \lambda \\
\end{bmatrix}

and B =

\begin{bmatrix}
\lambda & b \\
0 & \lambda \\
\end{bmatrix}

Assuming that a ≠ 0, and b ≠ 0 ; find a matrix X such that X-1AX = B.

2. Relevant equations

(A- $\lambda$I)v=0

3. The attempt at a solution

I tried using the following logic: Let B = {v1, v2,....vn}
be the basis of Fn consisting of the columns of X. We know that column j of B is
equal to [Avj ]B, that is, the coordinates of Avj with respect to the basis B.

But because of the two matrices having the exact same eigenvalues, I just end up with a=0, and am unable to actually find an invertible matrix X. Am I misreading the question, the logic, etc..?

2. Jan 24, 2012

### smerhej

Also in case the notation is different elsewhere, lambda = eigenvalue.

Last edited: Jan 24, 2012
3. Jan 24, 2012

### dirk_mec1

Am I missing something because from what I can tell just set up a 2x2 matrix with 4 unknown elements setup 4 equations and find a matrix X which satisfies those four equations.