# Identity matrix

In linear algebra, the **identity matrix** of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.

## Terminology and notation

The identity matrix is often denoted by , or simply by if the size is immaterial or can be trivially determined by the context.

The term **unit matrix** has also been widely used, but the term *identity matrix* is now standard. The term *unit matrix* is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all matrices.

In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, , or called "id" (short for identity). Less frequently, some mathematics books use or to represent the identity matrix, standing for "unit matrix" and the German word *Einheitsmatrix* respectively.

In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as

## Properties

When is an matrix, it is a property of matrix multiplication that

When matrices are used to represent linear transformations from an -dimensional vector space to itself, the identity matrix represents the identity function, for whatever basis was used in this representation.

The th column of an identity matrix is the unit vector , a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is .

The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:

- When multiplied by itself, the result is itself
- All of its rows and columns are linearly independent.

The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.

The rank of an identity matrix equals the size , i.e.:

## See also

- Binary matrix (zero-one matrix)
- Elementary matrix
- Exchange matrix
- Matrix of ones
- Pauli matrices (the identity matrix is the zeroth Pauli matrix)
- Householder transformation (the Householder matrix is built through the identity matrix)
- Square root of a 2 by 2 identity matrix
- Unitary matrix
- Zero matrix