Identity matrix

Not to be confused with matrix of ones, unitary matrix, or matrix unit.

In linear algebra, the identity matrix of size ${\displaystyle n}$ is the ${\displaystyle n\times n}$ 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 ${\displaystyle I_{n}}$, or simply by ${\displaystyle I}$ if the size is immaterial or can be trivially determined by the context.^[1]

$${\displaystyle I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \dots ,\ I_{n}={\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}.}$$

The term unit matrix has also been widely used,^[2][3][4][5] but the term identity matrix is now standard.^[6] 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 ${\displaystyle n\times n}$ matrices.^[7]

In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, ${\displaystyle \mathbf {1} }$, or called "id" (short for identity). Less frequently, some mathematics books use ${\displaystyle U}$ or ${\displaystyle E}$ to represent the identity matrix, standing for "unit matrix"^[2] and the German word Einheitsmatrix respectively.^[8]

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

$${\displaystyle I_{n}=\operatorname {diag} (1,1,\dots ,1).}$$

The identity matrix can also be written using the Kronecker delta notation:^[8]

$${\displaystyle (I_{n})_{ij}=\delta _{ij}.}$$

Properties

When ${\displaystyle A}$ is an ${\displaystyle m\times n}$ matrix, it is a property of matrix multiplication that

$${\displaystyle I_{m}A=AI_{n}=A.}$$

In particular, the identity matrix serves as the multiplicative identity of the matrix ring of all ${\displaystyle n\times n}$ matrices, and as the identity element of the general linear group ${\displaystyle GL(n)}$, which consists of all invertible ${\displaystyle n\times n}$ matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is an involutory matrix, equal to its own inverse. In this group, two square matrices have the identity matrix as their product exactly when they are the inverses of each other.

When ${\displaystyle n\times n}$ matrices are used to represent linear transformations from an ${\displaystyle n}$-dimensional vector space to itself, the identity matrix ${\displaystyle I_{n}}$ represents the identity function, for whatever basis was used in this representation.

The ${\displaystyle i}$th column of an identity matrix is the unit vector ${\displaystyle e_{i}}$, a vector whose ${\displaystyle i}$th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is ${\displaystyle n}$.

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

1. When multiplied by itself, the result is itself
2. 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.^[9]

The rank of an identity matrix ${\displaystyle I_{n}}$ equals the size ${\displaystyle n}$, i.e.:

$${\displaystyle \operatorname {rank} (I_{n})=n.}$$

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

Notes

1. "Identity matrix: intro to identity matrices (article)". Khan Academy. Retrieved 2020-08-14.
2. Pipes, Louis Albert (1963). Matrix Methods for Engineering. Prentice-Hall International Series in Applied Mathematics. Prentice-Hall. p. 91.
3. Roger Godement, Algebra, 1968.
4. ISO 80000-2:2009.
5. Ken Stroud, Engineering Mathematics, 2013.
6. ISO 80000-2:2019.
7. Weisstein, Eric W. "Unit Matrix". mathworld.wolfram.com. Retrieved 2021-05-05.
8. Weisstein, Eric W. "Identity Matrix". mathworld.wolfram.com. Retrieved 2020-08-14.
9. Mitchell, Douglas W. (November 2003). "87.57 Using Pythagorean triples to generate square roots of ${\displaystyle I_{2}}$". The Mathematical Gazette. 87 (510): 499–500. doi:10.1017/S0025557200173723. JSTOR 3621289.