Matrix Decomposition with restriction on eigenvalues

Question

Let $B$ be an invertible $n\times n$ complex matrix. Prove that there exist $n\times n$ complex matrices $A, C$ such that the following three conditions are satisfied simultaneously:

• (i) $B = AC$
• (ii) $A$ is diagonalizable, and $1$ is the only eigenvalue of $C$; and
• (iii) $AC=CA$

I am stumped by the condition that $1$ should be the only eigenvalue of $C$. How to find a matrix $C$ that fulfills this condition? Thanks for the help.

Answer 1

Hint. If $B$ is a single invertible Jordan block, we may choose $C=\frac1\lambda B$ and $A=\lambda I$.