In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' _{''n''}, or simply by ''I'' if the size is immaterial or can be trivially determined by the context.
: $I\_1\; =\; \backslash begin\; 1\; \backslash end\; ,\backslash \; I\_2\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; ,\backslash \; I\_3\; =\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \backslash end\; ,\backslash \; \backslash dots\; ,\backslash \; I\_n\; =\; \backslash begin\; 1\; \&\; 0\; \&\; 0\; \&\; \backslash cdots\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \&\; \backslash cdots\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \&\; \backslash cdots\; \&\; 0\; \backslash \backslash \; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; \backslash cdots\; \&\; 1\; \backslash end.$
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 _{n}'' represents the _{i}'' (the vector whose ''i''th entry is 1 and 0 elsewhere) It follows that the _{2}".

{{Matrix classes
Matrices
1 (number)
Sparse matrices

square matrix
In mathematics
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with ones on the main diagonal
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

and zeros elsewhere. It is denoted by ''I''matrix of onesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

and for any unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...

of the ring of all ''n''×''n'' matrices.
In some fields, such as group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

or quantum mechanics
Quantum mechanics is a fundamental theory
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...

, 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

that
: $I\_m\; A\; =\; A\; I\_n\; =\; A.$
In particular, the identity matrix serves as the multiplicative identity
In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...

of the ring of all ''n''×''n'' matrices, and as the identity element
In mathematics
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of the general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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 transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s from an ''n''-dimensional vector space to itself, ''Iidentity 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 (mathematics), function that always returns the same value that was ...

, regardless of the basis
Basis may refer to:
Finance and accounting
*Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items.
Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...

.
The ''i''th column of an identity matrix is the unit vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''edeterminant
In mathematics
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of the identity matrix is 1, and the trace
Trace may refer to:
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* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band)
Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...

is ''n''.
Using the notation that is sometimes used to concisely describe diagonal matrices
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is \left begin
3 & 0 \\
0 & 2 \end\right/math>, while ...

, we can write
: $I\_n\; =\; \backslash operatorname(1,\; 1,\; \backslash dots,\; 1).$
The identity matrix can also be written using the Kronecker delta
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

notation:
: $(I\_n)\_\; =\; \backslash delta\_.$
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
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

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
In the theory of vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

.
The principal square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of ...

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''The Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education
In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research
Researchers in mathem ...

87, November 2003, 499–500.
The rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

of an identity matrix equals the size ''n'', i.e.:
: $rank(I\_n)\; =\; n\; .$
See also

*Binary matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a with entries from the Such a matrix can be used to represent a between a pair of s.
Matrix representation of a relation
If ''R'' is a between the finite s ...

(zero-one matrix)
* Elementary matrixIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Exchange matrixIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

*Matrix of onesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Pauli matrices
In mathematical physics and mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...

(the identity matrix is the zeroth Pauli matrix)
*
* Unitary matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...

* Zero matrixIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

Notes