ELEMENTARY MATRIX Meaning and
Definition
-
An elementary matrix is a type of square matrix that represents a basic operation on another matrix. It is primarily utilized in linear algebra to simplify calculations and transformations on matrices. An elementary matrix is always obtained through a single elementary row operation or a single elementary column operation performed on an identity matrix.
In terms of row operations, an elementary matrix could be obtained by multiplying the identity matrix by another matrix resulting from any of the following operations: multiplying a row of the identity matrix by a non-zero scalar, swapping two rows of the identity matrix, or adding a scalar multiple of one row to another row of the identity matrix. Similarly, in terms of column operations, an elementary matrix could be derived by multiplying the identity matrix by another matrix resulting from any of the following operations: multiplying a column of the identity matrix by a non-zero scalar, swapping two columns of the identity matrix, or adding a scalar multiple of one column to another column of the identity matrix.
The elementary matrix concept allows for the efficient manipulation of matrices during computations and solving systems of linear equations. It is an essential tool in understanding matrix algebra and performing matrix operations with ease. By utilizing elementary matrices, complex procedures can be simplified into straightforward, step-by-step operations, facilitating a clear and systematic approach to linear transformations and matrix manipulations.
Etymology of ELEMENTARY MATRIX
The word "elementary" in the term "elementary matrix" derives from the mathematical concept of elementary operations. In linear algebra, elementary operations or elementary row operations are basic operations performed on a matrix to manipulate its rows. These operations include swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.
On the other hand, the word "matrix" comes from the Latin word "matrix" meaning "source" or "womb". It was originally used in mathematics to refer to an array of numbers or symbols used to represent a set of equations. The term "matrix" was first introduced by the English mathematician James Sylvester in the mid-19th century.
When combined, the term "elementary matrix" signifies a matrix that is the result of performing elementary row operations on the identity matrix.
