Exercise 2.1.2

For matrix $D$, we have $d_{i,i}=1$ for $i=1,\dots ,n$ and $d_{i,i-1}=-1$ for $i=2,\dots ,n$. For matrix $D^{-1}$, we have $d_{i,j}=1$ if $i\ge j$, otherwise $d_{i,j}=0$.

For the product, $M=D{D}^{-1}$, we have $M_{i,j}={\sum <!--FUNCTION\; APPLICATION-->}_k{d}_{i,k}{d}_{k,j}^{-1}=d_{i,i}{d}_{i,j}^{-1}+d_{i,i-1}{d}_{i-1,j}^{-1}=d_{i,j}^{-1}-d_{i-1,j}^{-1}$

• If $i<j$, we have $M_{i,j}=0-0=0$

• If $i=j$, we have $M_{i,j}=1-0=1$,

• If $i $\ge j+1\$,wehave$ Mi,j=\; 1\; -\; 1\; =\; 0$$

So $M=I=D{D}^{-1}$.

If $n=4$, we have $D^{-1}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$ so ${(D^{-1})}^T=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Then ${(D{D}^T)}^{-1}={(D^{-1})}^T{D}^{-1}=\left[\begin{array}{cccc}\hfill 4\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$