Identity matrix

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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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

The identity matrix is used in various applications, including:

Related Concepts[edit | edit source]

See Also[edit | edit source]

References[edit | edit source]

External Links[edit | edit source]

Contributors: Prab R. Tumpati, MD