Identity matrix

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In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.

Terminology and notation[edit]

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[edit]

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:

  1. When multiplied by itself, the result is itself
  2. 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[edit]

Notes[edit]

  1. ^ "Identity matrix: intro to identity matrices (article)". Khan Academy. Retrieved 2020-08-14.
  2. ^ a b Pipes, Louis Albert (1963). Matrix Methods for Engineering. Prentice-Hall International Series in Applied Mathematics. Prentice-Hall. p. 91.
  3. ^ Roger Godement, Algebra, 1968.
  4. ^ ISO 80000-2:2009.
  5. ^ Ken Stroud, Engineering Mathematics, 2013.
  6. ^ ISO 80000-2:2019.
  7. ^ Weisstein, Eric W. "Unit Matrix". mathworld.wolfram.com. Retrieved 2021-05-05.
  8. ^ a b Weisstein, Eric W. "Identity Matrix". mathworld.wolfram.com. Retrieved 2020-08-14.
  9. ^ Mitchell, Douglas W. (November 2003). "87.57 Using Pythagorean triples to generate square roots of ". The Mathematical Gazette. 87 (510): 499–500. doi:10.1017/S0025557200173723. JSTOR 3621289.