Definition of an Identity Matrix
An identity matrix, denoted by 'I' (or I_n for a specific dimension 'n'), is a square matrix where all elements along its main diagonal (from top-left to bottom-right) are 1, and all other elements are 0. It is a unique matrix for each dimension.
Key Properties
The defining property of an identity matrix is that when it is multiplied by any other compatible matrix (A), the original matrix remains unchanged (A * I = A and I * A = A). This makes it analogous to the number 1 in scalar multiplication or the additive identity (0) in addition.
Practical Example
For a 3x3 matrix, the identity matrix looks like: [[1, 0, 0], [0, 1, 0], [0, 0, 1]]. If you have a matrix A = [[2, 3, 4], [5, 6, 7], [8, 9, 0]], then A * I_3 will result in the original matrix A, and I_3 * A will also be A.
Importance and Applications
Identity matrices are crucial in linear algebra for tasks such as finding the inverse of a matrix, which is essential for solving systems of linear equations. They also represent transformations that leave an object exactly as it is (no change in position, size, or orientation) in computer graphics and physics.