Special Matrices

4. A diagonal matrix is a square matrix that has $0$s anywhere outside the main diagonal.
   • The elements on the diagonal could be anything, including any $0$s.
   • We could use the letter $D$ to denote a diagonal matrix. For example, $D = D_{2 \times 2} = D_2$.
     • For the diagonal matrix $D_n = [d_{i,j}]$, we have $d_{i,j} = 0$, where $1 \le i, j \le n$ but $i \neq j$.
     • We can also say:
       $D_n = \text{diag}(d_{1,1}, d_{2,2}, \dots, d_{n,n})$
       , where
       $[d_{1,1}, d_{2,2}, \dots, d_{n,n}]$
       are the diagonal elements.
       • Note:
         $D_n = \text{diag}(d_{1,1}, d_{2,2}, \dots, d_{n,n})$
         is a matrix of size $n \times n$, while $\text{diag}(D_n)$ is a one dimensional array ($1 \times n$) of the elements on the diagonal of $D_n$.
   • Singleton matrices and identity matrices are specific cases of diagonal matrices.
   Examples:

   3. $S_1 = \begin{bmatrix} \text{Bla} \end{bmatrix}$

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

   22. $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

   25. $L_3 = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$

   26. $Q_3 = \begin{bmatrix} -19 & 0 & 0 \\ 0 & 2.2 & 0 \\ 0 & 0 & \frac{1}{2} \end{bmatrix}$