Let $A\in M_{m,n}(\mathbb C)$ and $B\in M_{n,m}(\mathbb C)$ s.t $m\leq n$
Calculate the product $$\begin{pmatrix}
I_m & -A \\
0_{n,m} & I_n \\
\end{pmatrix}\begin{pmatrix}
AB & O_{m,n} \\
B & O_n \\
\end{pmatrix}\begin{pmatrix}
I_m & A \\
O_{n,m} & I_n \\
\end{pmatrix}$$
Deduce that the relation between $p_{AB}$ and $p_{BA}$ such that $p$ is characteristc polynomial.
And deduce that $\forall x,y \in \mathbb{C} ^n$ the characteristc polynomial of $xy^*$.
my attempt:
For the first question I find that: $$\begin{pmatrix}
I_m & -A \\
0_{n,m} & I_n \\
\end{pmatrix}\begin{pmatrix}
AB & O_{m,n} \\
B & O_n \\
\end{pmatrix}\begin{pmatrix}
I_m & A \\
O_{n,m} & I_n \\
\end{pmatrix}=\begin{pmatrix}
O_m & O_{m,n} \\
B & BA\\
\end{pmatrix}$$
For the second question: Let $M=\begin{pmatrix}
O_m & O_{m,n} \\
B & BA\\
\end{pmatrix}$ so we have $p_{M}:= det(XI_{m+n}-M)= det(X)p(BA)=X^mp(BA)$.
Can you help me please what is the characteristic polynomial of the left hand side.
You already obtained $p_M=X^mp(BA)$. Let $$ N=\begin{pmatrix} AB & O_{m,n} \\ B & O_n \\ \end{pmatrix} $$ Similarly, $p_{N}=\det(I_{m+n}X-N)=X^np(AB)$. Clearly $p_M=p_N$ so $$ X^np(AB)=X^mp(BA). \tag1 $$ Now for $A=x, B=y^*$ (suppose $x=(x_1,\cdots,x_n)^T$ which is a column vector or a $n\times1$ matrix and $y*=(y_1,\cdots,y_n)$ which is a row vector, or ($1\times n$) matrix), $$ BA=\sum_{k=1}^nx_ky_k $$ is a $1\times1$ matrix, and $AB$ is a $n\times n$ matrix. Using (1), one has $$ xp(AB)=X^np(BA)=X^n(X-\sum_{k=1}^nx_ky_k) $$ and hence $$ p(AB)=X^{n-1}(X-\sum_{k=1}^nx_ky_k). $$