# Show similarity of block matrices

Show that the following block matrices

$$P = \begin{bmatrix} AB & \mathrm O_{m \times n}\\B & \mathrm O_{n \times n}\end{bmatrix} \qquad \qquad Q = \begin{bmatrix} \mathrm O_{m \times m} & \mathrm O_{m \times n}\\B&BA\end{bmatrix}$$

where $A$ is an $m\times n$ and $B$ is an $n\times m$ matrix, are similar.

• But same eigenvalues does not guarantee similarity. Example: $$\begin {bmatrix}1&1\\0&1\end{bmatrix}$$ and $$\begin {bmatrix}1&0\\0&1\end{bmatrix}$$ Oct 27, 2016 at 21:21
• Ah, you are right. Sorry. Oct 27, 2016 at 21:24
• Here is an answer in this site: math.stackexchange.com/questions/427703/… Oct 27, 2016 at 21:27

$$\left[ \begin{array}{cr} 1_m & -A \\ 0 & 1_n\end{array}\right] \left[ \begin{array}{cr} AB & 0 \\ B & 0\end{array}\right] \left[ \begin{array}{cr} 1_m & A \\ 0 & 1_n\end{array}\right] = \left[ \begin{array}{cr} 0 & 0 \\ B & BA\end{array}\right]$$
It may be a little more illuminating to view the situation this way: $$\left[ \begin{array}{cr} 1_m & A \\ 0 & 1_n\end{array}\right] \left[ \begin{array}{cr} 0 & 0 \\ B & 0\end{array}\right] = \left[ \begin{array}{cr} AB & 0 \\ B & 0\end{array}\right]$$
$$\left[ \begin{array}{cr} 0 & 0 \\ B & 0\end{array}\right] \left[ \begin{array}{cr} 1_m & A \\ 0 & 1_n\end{array}\right] = \left[ \begin{array}{cr} 0 & 0 \\ B & BA\end{array}\right]$$