# Matrix 101 - Identity Matrix

Matrix is very important in understanding math in machine learning. I will spend a few posts to explain the basics and then focus on the applications in ML.

## symbols used in the post

Symbol | Meaning |
---|---|

$\mathbf{A}$ | the matrix, $\mathbf{A}_{mn}$ means a $m \times n $ matrix |

$A_{ij}$ | the $(i,j)$ element of the matrix |

$\mathbf{A}^{-1}$ | the inverse of matrix $\mathbf{A}$ |

$\mathbf{I}$ | the identity matrix, alternatively, use $\mathbf{I}_n$ means dimension $n$ identity matrix |

$\mathbf{0}$ | the zero matrix, or null matrix, all elements are 0 |

$\text{diag}$ | the diagonal of matrix |

## Definition

Identity matrix is a matrix where elements on the main or principal diagonal are 1 and all other elements are 0.

Alternatively, unit matrix is also used.

$$
\begin{bmatrix}
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{bmatrix}
$$

The math notation of a identity matrix can be described via a diagonal matrix $$ \mathbf{I}_n = \text{diag}(1, 1, 1, \cdots, 1) $$

## Properties

Identity matric has some good peroperties in term of matrix mulitplication

multiplying any matrix by the identity matrix results in the matrix itself

$$ \mathbf{I}_{m}\mathbf{A}_{mn} = \mathbf{A}_{mn} = \mathbf{A}_{mn}\mathbf{I}_{n} $$

Any matrix multiplied by its inverse reulsts in identity matrix

$$ \mathbf{A} \mathbf{A}^{-1} = \mathbf{I} $$

All the powers of identity matrix are equal to the identity matrix $$ \mathbf{I}^{n} = \mathbf{I} $$

The identity matrix is analogous to the number “1” in scalar algebra.