A lower triangular matrix is a square matrix where all the elements above the main diagonal are equal to 0. The main diagonal is the line that runs from the top left corner to the bottom right corner of the matrix.
In mathematics, a lower triangular matrix is commonly used for solving systems of linear equations, as it simplifies the process of solving the equations. It is also used in algorithms for solving optimization problems and in linear algebra operations such as matrix multiplication.
A lower triangular matrix has 0s above the main diagonal, while an upper triangular matrix has 0s below the main diagonal. Both matrices have non-zero elements on the main diagonal. Additionally, the elements in the lower triangular matrix are mirrored in the upper triangular matrix.
When storing a lower triangular matrix in memory, only the elements below the main diagonal are typically stored, as the elements above the diagonal are all equal to 0. The elements are usually stored in a 1-dimensional array, with the first element being the first row of the matrix, followed by the second row, and so on.
One advantage of using a lower triangular matrix is that it reduces the storage space needed, as only the non-zero elements are stored. It also simplifies calculations and operations, as the 0 elements can be ignored. Additionally, it can make solving systems of linear equations more efficient and less prone to errors.