# Identity matrix

The identity matrix is a square matrix with ones on the main diagonal and zeros in every other entry; multiplying any compatible matrix by it leaves that matrix unchanged, exactly as multiplying a number by 1 does.<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2019/references/ref-5-special-matrices/)</sup><sup> • </sup><sup>[2](https://sites.ualberta.ca/~jsylvest/books/DLA1/section-inverses-concepts.html)</sup> The identity matrix of order \( n \) is written \( I_n \), and when the size is clear from context simply \( I \).<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2019/references/ref-5-special-matrices/)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( I_n \) has all entries equal to zero except the diagonal, which is all 1; for example \( I_4 \) is the 4 × 4 matrix with 1s on its diagonal.<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2019/references/ref-5-special-matrices/)</sup> |
| Identity property | For any \( m \times n \) matrix \( A \), \( I_m A = A = A I_n \); \( I_m \) is a left identity and \( I_n \) a right identity.<sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup> |
| Uniqueness | \( I_n \) is the unique multiplicative identity for matrix multiplication in \( \mathbb{R}^{n \times n} \).<sup>[4](https://people.wou.edu/~beavers/Text341/Book%20Updated%2010-5-22/sectionMatrixAlgebra.html)</sup> |
| Kronecker delta form | \( (I_n)_{ij} = \delta_{ij} \), where \( \delta_{ij} \) equals 1 when \( i = j \) and 0 otherwise.<sup>[5](https://planetmath.org/kroneckerdelta)</sup><sup> • </sup><sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup> |
| Geometric meaning | The identity matrix represents the transformation that leaves every vector where it is.<sup>[6](https://www.learnmathclass.com/linear-algebra/visual-tools/matrix-types)</sup> |
| Conditioning | \( \operatorname{cond}(I) = 1 \), the smallest possible condition number for a matrix.<sup>[7](https://cs357.cs.illinois.edu/textbook/notes/condition.html)</sup> |
| Powers | For a square matrix \( A \) of order \( n \), \( A^0 = I_n \).<sup>[8](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/matrices.html)</sup> |

## Definition and notation

A square matrix with ones on the main diagonal and zeros everywhere else is called an identity matrix.<sup>[9](https://sites.math.duke.edu/education/ccp/materials/engin/matrix/matr3.html)</sup> The \( n \times n \) identity matrix is denoted \( I_n \):<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2019/references/ref-5-special-matrices/)</sup>

\[ I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} \]

A compact way to define it uses the [Kronecker delta](https://www.edgepedia.org/kronecker-delta) \( \delta_{ij} \), defined as 1 if \( i = j \) and 0 if \( i \neq j \). The identity matrix is simply the matrix of the delta, \( (I_n)_{ij} = \delta_{ij} \).<sup>[5](https://planetmath.org/kroneckerdelta)</sup><sup> • </sup><sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup> The delta itself entered the literature through Luther Pfahler Eisenhart's 1926 book *Riemannian Geometry*, after being used in lectures by Leopold Kronecker (1823–1891).<sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup>

Because the identity matrix is square, it exists only in square form; there are no non-square identity matrices.<sup>[10](https://apprize.best/programming/opengl_1/6.html)</sup>

## Multiplicative identity

The defining property is that the identity matrix acts as a multiplicative identity. For any \( m \times n \) matrix \( A \),<sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup>

\[ I_m A = A = A I_n. \]

The two sides use different identities: \( I_m \) is a left multiplicative identity and \( I_n \) a right one, and the sizes must match the multiplication. When \( A \) is square, \( I_n \) serves on both sides and commutes with \( A \); in fact any scalar multiple of \( I \) commutes with \( A \), though other commuting matrices may also exist.<sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup>

The proof of the left identity property follows from the delta structure. Writing \( B = I_m A \), the entry \( b_{ij} = \sum_{k=1}^{m} \delta_{ik} a_{kj} = a_{ij} \), since every term with \( i \neq k \) vanishes; hence \( B = A \), and a similar argument gives the right-hand equality.<sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup>

**Uniqueness.** Within \( \mathbb{R}^{n \times n} \), \( I_n \) is the unique matrix satisfying \( AI = IA = A \) for all \( A \) in the space; it also fixes every vector, \( I b = b \).<sup>[4](https://people.wou.edu/~beavers/Text341/Book%20Updated%2010-5-22/sectionMatrixAlgebra.html)</sup> More generally, in the set of \( n \times n \) matrices with entries in a unit ring, the identity matrix is the multiplicative identity, and it is also the identity of the general linear group \( GL(n,K) \) over a field \( K \) and of all its subgroups.<sup>[11](https://mathworld.wolfram.com/MultiplicativeIdentity.html)</sup> Not every multiplicative structure contains it: the set of \( n \times n \) matrices with determinant zero is closed under multiplication but does not include the identity matrix.<sup>[11](https://mathworld.wolfram.com/MultiplicativeIdentity.html)</sup>

The identity matrix plays the role for multiplication that the zero matrix plays for addition, and it is what makes the notion of a multiplicative inverse meaningful for matrices.<sup>[2](https://sites.ualberta.ca/~jsylvest/books/DLA1/section-inverses-concepts.html)</sup> A square matrix \( A \) is invertible (or nonsingular) when there exists \( B \) with \( AB = BA = I_n \), and \( B \) is then written \( A^{-1} \).<sup>[12](https://www.lancaster.ac.uk/~prendivs/accessible/math105/lecturenotes.tex/S3.SS1.html)</sup> Only square matrices can possibly be invertible.<sup>[12](https://www.lancaster.ac.uk/~prendivs/accessible/math105/lecturenotes.tex/S3.SS1.html)</sup> One wrinkle distinguishes matrix algebra from ordinary number algebra: while any matrix can be canceled to the zero matrix by subtraction, a matrix cannot always be canceled to the identity by dividing, because an inverse need not exist.<sup>[2](https://sites.ualberta.ca/~jsylvest/books/DLA1/section-inverses-concepts.html)</sup>

## As a linear transformation

Every matrix encodes a linear transformation, and the identity matrix is the matrix of the transformation that leaves every vector where it is.<sup>[6](https://www.learnmathclass.com/linear-algebra/visual-tools/matrix-types)</sup> Multiplying a vector by \( I \) returns the vector unchanged: each output component is a sum in which only the diagonal 1 contributes, so \( 1 \cdot x_1 + 0 \cdot x_2 + \cdots = x_1 \), and similarly for the other components.<sup>[13](https://learnopengl.com/Transformations)</sup>

This no-op role makes \( I \) a practical starting point in computation. In computer graphics, transformation matrices are built by beginning with the identity and composing scaling, rotation, and translation operations onto it; in OpenGL-style code a 4 × 4 matrix is commonly initialized to the identity before transformations are applied, since starting from an all-zero matrix would make every subsequent operation produce zeros.<sup>[13](https://learnopengl.com/Transformations)</sup><sup> • </sup><sup>[14](https://learnopengl.com/Getting-started/Transformations)</sup> Objects drawn under the identity matrix appear untransformed, at the origin with axes matching eye coordinates.<sup>[10](https://apprize.best/programming/opengl_1/6.html)</sup>

## Algebraic properties

Several standard facts follow directly from the definition:

- **Commutation.** \( I \) commutes with every square matrix of matching size, one of the few guaranteed instances of commutativity in matrix multiplication, which otherwise generally fails.<sup>[3](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)</sup><sup> • </sup><sup>[8](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/matrices.html)</sup>
- **Powers.** For a square matrix \( A \), powers are defined by repeated multiplication, with \( A^0 = I_n \), \( A^2 = AA \), and so on.<sup>[8](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/matrices.html)</sup>
- **Transpose.** Any identity matrix is its own transpose.<sup>[10](https://apprize.best/programming/opengl_1/6.html)</sup>
- **Rank.** The rank of an \( n \times n \) identity matrix is \( n \).<sup>[15](https://bobson.ludost.net/books/algo/book6/chap31.htm)</sup>
- **Conditioning.** For any matrix, \( \operatorname{cond}(A) = \|A\| \, \|A^{-1}\| \geq \|AA^{-1}\| = \|I\| = 1 \), so \( \operatorname{cond}(I) = 1 \) is the smallest possible condition number; the identity matrix is well conditioned, and orthogonal matrices share this optimal conditioning.<sup>[7](https://cs357.cs.illinois.edu/textbook/notes/condition.html)</sup>

The identity matrix is also a diagonal matrix (all off-diagonal entries zero) and a permutation matrix (a single 1 in each row and column), making it the trivial member of both families.<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2019/references/ref-5-special-matrices/)</sup><sup> • </sup><sup>[16](https://coral.ise.lehigh.edu/~ted/files/ie170/lectures/Lecture26.pdf)</sup>

## Notation and history

[Arthur Cayley](https://www.edgepedia.org/arthur-cayley), who gave the first abstract definition of a matrix in his 1858 *Memoir on the theory of matrices*, wrote the identity matrix as 1 and the zero matrix as 0; both [Joseph Wedderburn](https://www.edgepedia.org/joseph-wedderburn) in *Lectures on Matrices* (1934) and MacDuffee used \( I \) and \( O \).<sup>[17](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/matrices/)</sup><sup> • </sup><sup>[18](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup> The modern convention of subscripting by size, \( I_n \), keeps the order explicit when several matrix sizes are in play, and many texts drop the subscript when the context fixes it.<sup>[4](https://people.wou.edu/~beavers/Text341/Book%20Updated%2010-5-22/sectionMatrixAlgebra.html)</sup><sup> • </sup><sup>[8](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/matrices.html)</sup>

## References

1. [Special Matrices, CS 357, University of Illinois](https://courses.grainger.illinois.edu/cs357/fa2019/references/ref-5-special-matrices/)
2. [Section 5.3 Concepts: Inverses, DLA Concepts, University of Alberta](https://sites.ualberta.ca/~jsylvest/books/DLA1/section-inverses-concepts.html)
3. [8. Matrix Algebra, MATH METHODS (FIU)](https://faculty.fiu.edu/~boydj/mathecon/math08.pdf)
4. [Matrix Algebra, Western Oregon University textbook](https://people.wou.edu/~beavers/Text341/Book%20Updated%2010-5-22/sectionMatrixAlgebra.html)
5. [Kronecker delta, PlanetMath](https://planetmath.org/kroneckerdelta)
6. [Special Matrices, Learn Math Class](https://www.learnmathclass.com/linear-algebra/visual-tools/matrix-types)
7. [Condition Numbers, CS 357 Textbook, University of Illinois](https://cs357.cs.illinois.edu/textbook/notes/condition.html)
8. [3.3. Matrices, Discrete Structures for Computing, Western University](https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter3/matrices.html)
9. [Matrix Operations, Part 3, Duke University](https://sites.math.duke.edu/education/ccp/materials/engin/matrix/matr3.html)
10. [Math for 3D Graphics, OpenGL SuperBible, Sixth Edition](https://apprize.best/programming/opengl_1/6.html)
11. [Multiplicative Identity, Wolfram MathWorld](https://mathworld.wolfram.com/MultiplicativeIdentity.html)
12. [3.1 Identity matrices and inverses, MATH 105, Lancaster University](https://www.lancaster.ac.uk/~prendivs/accessible/math105/lecturenotes.tex/S3.SS1.html)
13. [LearnOpenGL, Transformations](https://learnopengl.com/Transformations)
14. [LearnOpenGL, Getting started: Transformations](https://learnopengl.com/Getting-started/Transformations)
15. [Introduction to Algorithms, Chapter 31: Matrix Operations](https://bobson.ludost.net/books/algo/book6/chap31.htm)
16. [IE170 Lecture 26, Lehigh University](https://coral.ise.lehigh.edu/~ted/files/ie170/lectures/Lecture26.pdf)
17. [Earliest Uses of Symbols for Matrices and Vectors, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Miller/mathsym/matrices/)
18. [Matrices and determinants, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Matrix theory › Structured and special matrix classes*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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