identity matrix

In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by ''I''_''n'', or simply by ''I'' if the size is immaterial or can be trivially determined by the context.

: $I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end.$

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 ''n''×''n'' matrices.

In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, 1, or called "id" (short for identity); otherwise it is identical to ''I''. Less frequently, some mathematics books use ''U'' or ''E'' to represent the identity matrix, meaning "unit matrix" and the German word respectively.

When ''A'' is ''m''×''n'', it is a property of

matrix multiplication

that
: $I_m A = A I_n = A.$
In particular, the identity matrix serves as the multiplicative identity of the ring of all ''n''×''n'' matrices, and as the identity element of the general linear group GL(''n'') (a group consisting of all invertible ''n''×''n'' matrices). In particular, the identity matrix is invertible—with its inverse being precisely itself.

Where ''n''×''n'' matrices are used to represent linear transformations from an ''n''-dimensional vector space to itself, ''I_n'' represents the

identity function

, regardless of the basis.

The ''i''th column of an identity matrix is the unit vector ''e_i'' (the vector whose ''i''th entry is 1 and 0 elsewhere) It follows that the

determinant

of the identity matrix is 1, and the trace is ''n''.

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write
: $I_n = \operatorname(1, 1, \dots, 1).$

The identity matrix can also be written using the Kronecker delta notation:
: $(I_n)_ = \delta_.$

When the identity matrix is the product of two square matrices, the two matrices are said to be the inverse of each other.

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.Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of ''I''₂". The Mathematical Gazette 87, November 2003, 499–500.

The rank of an identity matrix equals the size ''n'', i.e.:
: $rank(I_n) = n .$

See also

* Binary matrix (zero-one matrix)
* Elementary matrix
* Exchange matrix
*Matrix of ones
* Pauli matrices (the identity matrix is the zeroth Pauli matrix)
*

Square root of a 2 by 2 identity matrix

* Unitary matrix
* Zero matrix

Notes

{{Matrix classes

Matrices
1 (number)
Sparse matrices