Special Matrices

8. A dense matrix is a matrix that has a relatively small number of zero elements in it.
   • 'Dense' is the opposite of 'sparse': a sparse matrix is NOT dense, and vice versa.
   • The density ratio for a matrix $A_{n \times m}$ is the following number: $\text{density}(A) =$$\; \frac{\text{number of nonzero elements}}{n \cdot m}$.
     • Note that: $\text{density}(A) =$$\; 1 - \text{sparsity}(A)$.
     • For $A$ to be considered dense in our course, the following must be true: $\text{density}(A) > 0.5$.
   Examples of dense matrices:

   19. $I_1 = \begin{bmatrix} 1 \end{bmatrix}$

   14. $M_3 = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 4 \end{bmatrix}$

   5. $T = \begin{bmatrix} 65.7 & 62.4 & 73.8 & 76.6 & 75.0 \end{bmatrix}$

   36. $B = \begin{bmatrix} 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \end{bmatrix}$