In linear algebra, the identity matrix of size
is the
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with
one
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
s on the
main diagonal and
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
s elsewhere.
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
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
or
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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 respectively.
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:
Properties
When
is an
matrix, it is a property of
matrix multiplication that
In particular, the identity matrix serves as the
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
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
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
, for whatever
basis was used in this representation.
The
th column of an identity matrix is the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
, a vector whose
th entry is 1 and 0 elsewhere. The
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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
Notes
{{Matrix classes
Matrices
1 (number)
Sparse matrices