Identity matrix

Identity matrix

An identity matrix is a special type of square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. It is denoted by the symbol I or sometimes In to indicate its size. The identity matrix plays a crucial role in various areas of linear algebra and matrix theory.

Definition

An identity matrix of size n × n is defined as: \[ I_n = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix} \]

Properties

The identity matrix has several important properties:

• **Multiplicative Identity**: For any matrix A of size m × n, multiplying by the identity matrix does not change A. That is, \( A \cdot I_n = A \) and \( I_m \cdot A = A \).
• **Invertibility**: The identity matrix is its own inverse matrix, meaning \( I_n^{-1} = I_n \).
• **Determinant**: The determinant of the identity matrix is always 1, regardless of its size.
• **Eigenvalues**: All the eigenvalues of the identity matrix are 1.

Applications

The identity matrix is used in various applications, including:

• **Solving Linear Systems**: It is used in Gaussian elimination and other methods for solving systems of linear equations.
• **Matrix Decomposition**: It appears in matrix decomposition techniques such as LU decomposition and QR decomposition.
• **Transformations**: In computer graphics, the identity matrix is used as the starting point for transformation matrices.

Related Concepts

• Diagonal matrix
• Inverse matrix
• Orthogonal matrix
• Unit matrix

See Also

• Matrix (mathematics)
• Linear algebra
• Matrix multiplication
• Determinant
• Eigenvalue