characteristic polynomial of matrix with several blocks

Question

$\mathbf{M}=\begin{bmatrix} \mathbf{y}&0&0&\cdots&0&0&\mathbf{x}\\ \mathbf{I}&0&0&\cdots&0&0&0\\ 0&\mathbf{I}&0&0&0&0&0\\ 0&0&\mathbf{I}&0&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&\mathbf{I}&0&0\\ 0&0&0&0&0&\mathbf{I}&0\\ \end{bmatrix}$

Given that $\mathbf{y}=\operatorname{diag}\left(y_i\right)$ and $\mathbf{x}=\operatorname{diag}\left(x_i\right) $ where each $i=1,2,\dots, N$, $y_i$ are $N\times N$ matrices whose eigenvalues are strictly less inside unit circle. $I$ are $N\times N$. For each $i$, $x_i$ are also $N\times N$ Could anyone tell me what would be the characterstic polynomial of $\mathbf{M}$? Thanks.

I was trying the technique a block matrix proof about characteristic polynomials but could not proceed much.

Also, does taking the transpose of $M$ help?

I can't make the dimensions work. You say the $I$ are $N\times N$. So $y$ has $N$ columns. But you also say $y$ is diagonal with $n$ blocks of size $N\times N$ along the diagonal. Can you clarify? — ancient mathematician, Feb 16 '21 at 16:21
I still don't understand your question. As written, if $I$ is $N\times N$ then $y$ has $N$ columns. You then say that $y$ is a diagonal of $N$ blocks $y_i$. This forces the $y_i$ to have $1$ column each. But then you say $y_i$ is $N\times N$. So $N$ must be $1$. I'm sure you don't mean that? So please edit the question again. — ancient mathematician, Feb 17 '21 at 08:06
Come one each $y_i$ is $N\times N$ matrix so $\mathbf{y}$ is a block diagonal matrix right? — Myshkin, Feb 17 '21 at 08:59
Tell me: what are the dimensions of (i) the matrix $y$ (ii) the matrix $y_1$? — ancient mathematician, Feb 17 '21 at 10:19
So if $y$ is $N^2\times N^2$ how big is the $I$ just below it? It can't be $N\times N$ unless $N=1$> — ancient mathematician, Feb 17 '21 at 10:41
Look: you write this as a big block matrix, and now you tell me that the blocks you show don't match in size. As I keep saying $I$ and $y$ must have the same number of columns. — ancient mathematician, Feb 17 '21 at 10:47
If you make the $I$ and the $0$ $N^2 \times N^2$ then the question makes sense. (Though it looks hard). You should also specify the size of $M$, something like $(k+1)N^2$. — ancient mathematician, Feb 17 '21 at 12:19
On the other hand, what would be a simpler dimension which you think can solve the problem? Thanks so much for your help — Myshkin, Feb 17 '21 at 12:47
At present I can do $N=1,k=0$ and not much more. ;-) I will think. — ancient mathematician, Feb 17 '21 at 12:53

characteristic polynomial of matrix with several blocks

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