Identity Matrix

TL;DR

• A square matrix that leaves vectors unchanged when used as a linear transformation.
• Acts as the neutral element for matrix multiplication.
• Commonly denoted by the symbol “I” or “1”.

Definition

An identity matrix is a square matrix with ones on the main diagonal and zeros everywhere else. It is also called an identity element or unit matrix and is denoted by the symbol “I” or “1”.

Explanation

• The identity matrix is always square, meaning it has the same number of rows and columns.
• Its entries on the main diagonal are ones; all other entries are zeros.
• In linear algebra, it represents the identity transformation, which leaves an object unchanged.
• In matrix multiplication, the identity matrix is the neutral element: multiplying any conformable matrix by the identity matrix does not change that matrix.
• In computer science, the identity matrix is used in areas such as image processing and computer graphics to reset or represent no transformation.

Examples

Example 1:

$$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Example 2:

$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

Use cases

• Representing the identity transformation in linear algebra.
• Serving as the neutral element in matrix multiplication.
• Resetting transformations in image processing and computer graphics.

Related terms

• identity element
• unit matrix
• main diagonal
• symbols: “I” or “1”