We have defined the matrix X via the equations
y0=β0x00+β1x01+β2x02+⋯+βn−1x0n−1+ϵ0y1=β0x10+β1x11+β2x12+⋯+βn−1x1n−1+ϵ1y2=β0x20+β1x21+β2x22+⋯+βn−1x2n−1+ϵ1……yi=β0xi0+β1xi1+β2xi2+⋯+βn−1xin−1+ϵ1……yn−1=β0xn−1,0+β1xn−1,2+β2xn−1,2+⋯+βn−1xn−1,n−1+ϵn−1.As we noted above, we stayed with a system with the design matrix X∈Rn×n, that is we have p=n. For reasons to come later (algorithmic arguments) we will hereafter define our matrix as X∈Rn×p, with the predictors refering to the column numbers and the entries n being the row elements.