ABSTRACT
Matrices usually represent systems of equations. Matrix operations go beyond cross products and adding vectors. Most of the interesting applications of matrices require the three elementary row operations. This chapter explores a suite of tools for manipulating and transforming matrices. In the single-band diagonal matrix, the matrix can be viewed as the result of a nearly completed Gaussian elimination. The same is true of a tridiagonal matrix, except it is not nearly as complete. LU decomposition offers a better way of solving systems of equations using matrices. Like the LU decomposition, the Cholesky decomposition can be used to solve matrix equations. Both the Jacobi and Gauss–Seidel iterative methods provide some advantages over other methods for solving a matrix. There are numerous applications of linear algebra and solutions to matrices ranging from engineering, to physics, to statistical analysis. It might be simpler to list disciplines not using linear algebra, implicitly or explicitly.
