Square matrix with ones on the main diagonal and zeros elsewhere
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.
[1]
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 matrices.
[7]
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"
[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
The identity matrix can also be written using the
Kronecker delta notation:
[8]
Properties
When is an matrix, it is a property of
matrix multiplication that
In particular, the identity matrix serves as the
multiplicative identity of the
matrix ring of all
matrices, and as the
identity element of the
general linear group , which consists of all
invertible 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 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.
[9]
The
rank of an identity matrix equals the size , i.e.:
See also
Notes