Special Matrices

5. An upper triangular matrix is a square matrix that has $0$s anywhere below the main diagonal.
   • The elements on the diagonal and above it could be anything, including any $0$s.
   • We could use the letter $U$ to denote an upper triangular matrix.
     • For the upper triangular matrix $U_n = [u_{i,j}]$, we have $u_{i,j} = 0$, where $1 \le i, j \le n$ and $i > j$.
   • A diagonal matrix is a specific case of an upper triangular matrix.
   Examples:

   22. $E_1 = \begin{bmatrix} 🗽 \end{bmatrix}$

   28. $U_2 = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$

   29. $U_3 = \begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix}$

   30. $U_4 = \begin{bmatrix} -19 & 0 & 5 & 4e3 \\ 0 & 1.11 & 7 & 0 \\ 0 & 0 & 2^{100} & \frac{1}{2} \\ 0 & 0 & 0 & 3\end{bmatrix}$