A Toeplitz matrix, or diagonal$-$constant matrix, is a matrix in which each descending diagonal element

from left to right is constant. For example, the following $nn$ matrix is a Toeplitz matrix:

Using NumPy and some of the fundamental Python data structures we have been discussing in class, instantiate

in a Jupyter Notebook cell the $33$ Toeplitz matrix $K_3,$

$K_3=\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 2\end{array}\right]$

From $K_3,$ instantiate $T_3$ where only the upper$-$right corner element $($i$.$e$.,$ the top $a_0$ entry at location $1,1)$ is set to

$1,$

$T_3=\left[\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 2\end{array}\right]$

Using Python NumPy and data structure operations, perform two row operations on $T_3$ to define new matrix $U_3$

$R2$longleftarrowR$2+R1$

$R3$longleftarrowR$3+R2$

That is$,$ row $2$ is assigned row $2$ plus row $1,$ and row $3$ is assigned row $3$ plus row $2.$ Verify matrix $U_3$ is upper$-$

triangular Toeplitz with $1s$ on the main diagonal and $-1s$ on the diagonal just above the main.

In this assignment, you will explore how $T^{-1}$ can be computed from $U.$

A$.$ In a Jupyter Notebook cell, verify $T_3=U_3^T{U}_3.$

B$.$ In another cell, verify $U_3{U}_3^{-1}=I_3.$

C$.$ In a third cell, verify $(U_3^T{U}_3{)}^{-1}=T_3^{-1}.$ Note, $(U^{T}U{)}^{-1}=(U^{-1})(U^{-1}{)}^T.$

Finally, complete steps A$,$ B$,$ and C with $K_4,$

$K_4=\left[\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right]$