1. They arise in solving matrix equations such as the Sylvester equation. 2. Substituting the approximation into boundary condition yields the following matrix equation 3. The relation between the incident and scattered waves is given by the matrix equation 4. A matrix representation for an inhomogeneous medium was presented using a pair of matrix equations. 5. With multiple state variables and multiple control variables, the Riccati equation will be a matrix equation. 6. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. 7. :This matrix equation is solvable because the determinant is non-zero by Vandermonde's formula. 8. And this matrix equation contains the general conditions on the Lorentz transformation to ensure invariance of the spacetime interval. 9. Any vector- matrix equation may be transposed to an equivalent equation where the order of the factors is reversed. 10. I need to solve thousands of such matrix equations where A _ i typically has tens of thousands of rows.