Matrisenheter

Matrisenheter, also known as matrix units or elementary matrices, are fundamental building blocks in linear algebra. A matrisenhet is a special type of matrix that contains a single entry equal to 1 and all other entries equal to 0. For a given size n x n, there are n^2 distinct matrisenheter. Typically, a matrisenhet is denoted by $E_{ij}$, where the subscript 'i' indicates the row number and the subscript 'j' indicates the column number of the non-zero entry. Thus, the entry in the i-th row and j-th column of $E_{ij}$ is 1, and all other entries are 0.

These matrices have important properties related to matrix multiplication. When a matrix A is multiplied by a matrisenhet $E_{ij}$ on the left (i.e., $A \cdot E_{ij}$), the result is a matrix where the j-th column of A is moved to the i-th column, and all other columns are zeroed out. Conversely, multiplying a matrix A by $E_{ij}$ on the right (i.e., $E_{ij} \cdot A$) results in a matrix where the i-th row of A is moved to the j-th row, and all other rows are zeroed out.

Matrisenheter are crucial for understanding matrix operations, proving theorems about matrices, and are used in various applications such as solving systems of linear equations, basis representations, and understanding the structure of linear transformations. They can be viewed as matrices that perform elementary row or column operations.