I am messing around with signal processing, and I figured out the equation of a delay feedback loop, but I want to write it out in linear algebra. The problem is that I'm not quite sure how to format the linear algebra to take into account the recursion. I've done linear algebra in the past, but haven't studied any recursive methodologies, although it seems like if I can understand this case, I will understand other cases. $x[n]$ is an input signal, and $y[n]$ is an output signal. $D$ is a delay operator by 1 index. The barebones of the equation is as follows. $$ y[n]=x[n]+ D\lbrace y[n] \rbrace=x[n]+y[n-1], \space \space y[0]=x[0]=1$$
I have an input signal vector that is a essentially a unit-pulse, or dirac-delta pulse at $x_0$ of my sign. Every finite length signal of length $N$ is equivalent to a vector in $\mathbb{C}^{N}$, so let's make our input signal: $$ x =[x_0 \space x_1 \space x_2 \space x_3]^T=[1 \space 0 \space 0 \space 0]^T$$
I am going to send this input signal to an output signal $y$. On the way to $y$ it enters a loop back to itself and encounters a delay, which essentially has the effect of an operator matrix, we'll call $D$. The effect of $D$ on some signal z is: $$ Dx=D[x_0 \space x_1 \space x_2 \space x_3]^T = [x_3 \space x_0 \space x_1 \space x_2]^T $$ In our case in $\mathbb{C}^4$, $$ D= \left[ {\begin{array}{cccc} 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{array} } \right] $$
I'm confused as to how to use this information to reformat the equation into matrix/vector form. All I have so far is:
$$ y=x+Dy $$
However, I don't know if this is correct, and it doesn't take into account the initial conditions imposed onto my original equation that $y[0]=x[0]=1$. Since there is only 1 delay operator delaying by 1 index, I believe $y$ should equal $[1 \space 1 \space 1 \space 1]^T$ .
I apologize if my question isn't concise enough, or formatted properly. This is my first question.