Confusion with differential equation applied as a matrix

Question

http://i.imgur.com/PknZsfZ.png

 

I'm confused on why the first row has values 0 + y_k+1 + 0 and the second row has 0 + 0 + y_k+2

Answer 1

In the bottom row of the image (formula for x_k+1), the first term after the equals sign is the definition of x_k+1 based on the previously defined x_k (notice that all the k indices are just pushed forward by one). The next term after that is still a 3x1 vector. Each row entry is the corresponding element of x_k+1 written out as a linear combination of y_k, y_k+1, and y_k+2. 

 

The next step pulls out the y-elements as the original x_k vector, and leaves the coefficients behind as part of a 3x3 matrix. 

 

Overall this lets you write: x_k+1 = Mx_k, where M is that matrix of coefficients. In the context of the problem, this is important because now all you have to do to step your x vector forward (any number of times) is multiply by the coefficient matrix a corresponding number of times (e.g. x_k+4=MMMMx_k)