Identity matrix
Definition
An identity matrix, denoted by I, is a special type of square matrix that has 1s on the main diagonal and 0s in all its off-diagonal entries. The main diagonal is the set of entries A[i][i] where the row number and column number are the same.
Properties
Identity matrices have several unique properties that distinguish them from other matrices:
- Multiplicative Identity: For any given square matrix A, the product of A and the identity matrix is always A, regardless of the order of multiplication. That is, AI = IA = A. This property is why the matrix is called an "identity" matrix, as it behaves similarly to the multiplicative identity number 1 in basic arithmetic.
- Additive Identity: If an identity matrix is subtracted from a square matrix of the same order, the result is the same matrix. That is, A - I = A.
- Determinant: The determinant of an identity matrix is always 1, regardless of its order. The determinant is a special number that can be calculated from a matrix.
- Inverse: The inverse of an identity matrix is the identity matrix itself. That is, I-1 = I. The inverse of a matrix A is the matrix that, when multiplied with A, results in the identity matrix.
- Transpose: The transpose of an identity matrix is the identity matrix itself. The transpose of a matrix is obtained by interchanging its rows into columns or columns into rows.
Applications
Identity matrices are used in a wide range of mathematical and scientific fields, including linear algebra, computer science, and engineering.
In linear algebra, they are used in the definition of linear transformations and to represent the identity transformation. In computer science, they are used in graphics transformations, such as scaling, rotation, and translation. In engineering, they are used in control theory and signal processing.