Elementary Matrices

Elementary Matrices can be obtained from the Identity Matrix by an Elementary Row Operation. They are denoted with where denotes the row of the non-zero value and denotes the column.

Examples:

For LU Decomposition we focus on elementary matrices that represent the elementary row operations of the type ”add a scalar multiple of one row to another row”. In particular describes an Elementary matrix with the operation "multiply row with and add to row " this matrix looks like:

Example:
Perform row operation to the matrix A:

Calculating the matrix-matrix multiplication we get the result:

Note: This is the same as applying the row operation .