Please help me understand the proof below.
Here are my questions:
1.) what does r$_1$ represent?
2.) is $A_1$ just matrix $A$ with the first row and first column deleted?
3.) what does $\tilde{r_1}$ represent?
4.) why does r$_1$$Y$=[r$_{12}$$\tilde{r_1}$]
I'm sorry if this is a difficult question to answer. I can't tell if I am confused by some notational stuff or if I am not understanding some concepts. Please help me arrive at an understanding of this proof.
Regarding 1,2: in the equations $$ \begin{align} AX &= \cdots = \pmatrix{\lambda \mathbf x_1 & A \mathbf x_2 & \cdots & A \mathbf x_n} \\ & = \pmatrix{\mathbf x_1 & \mathbf x_2 & \cdots & \mathbf x_n} \pmatrix{\lambda_1 & \mathbf r_1 \\ 0 & A_1} = \cdots, \end{align} $$ the point being made is that there exists some matrix $M$ for which $AX = [\mathbf x_1 \ \ \cdots \ \ \mathbf x_n] M$, and this $M$ must have the first column $[\lambda_1,0,\dots,0].$ An efficient way to convey this is to avoid introducing the matrix $M$ and instead use the block-partitioned matrix $$ M = \pmatrix{\lambda_1 & \mathbf r_1 \\ 0 & A_1}. $$ So, to answer your question, $\mathbf r_1$ is a $1 \times (n-1)$ row-vector and $A_1$ is an $(n - 1) \times (n-1)$ matrix. Both of these are submatrices of the (unique) matrix $M$ satisfying $AX = XM$.
Regarding 3,4, we have the same thing going on. The author would like to break up the $1 \times (n-1)$ matrix $\mathbf r_1 Y$ into two submatrices. $r_{12}$ is the first entry of $\mathbf r_1 Y$, and $\tilde {\mathbf r}_1$ is the $1 \times (n-2)$ matrix built from the remaining entries of $\mathbf r_1 Y$.