Identity matrix

In linear algebra, the identity matrix (sometimes ambiguously called a unit matrix) of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context.[1][2] In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I. Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix"[3] and the German word Einheitsmatrix respectively.[4]

A 3 (row) x3 (column) Identity matrix of 1s on the diagonal, with 0s elsewhere
3x3 Identity matrix

When A is m×n, it is a property of matrix multiplication that

In particular, the identity matrix serves as the unit of the ring of all n×n matrices, and as the identity element of the general linear group 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 transformations from an n-dimensional vector space to itself, In represents the identity function, regardless of the basis.

The ith column of an identity matrix is the unit vector ei (the vector whose ith entry is 1 and 0 elsewhere) It follows that the determinant of the identity matrix is 1, and the trace is n.

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write

The identity matrix can also be written using the Kronecker delta notation:[4]

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 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.[5]

See alsoEdit

NotesEdit

  1. ^ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-14.
  2. ^ "Identity matrix: intro to identity matrices (article)". Khan Academy. Retrieved 2020-08-14.
  3. ^ Pipes, Louis Albert (1963). Matrix Methods for Engineering. Prentice-Hall International Series in Applied Mathematics. Prentice-Hall. p. 91.
  4. ^ a b Weisstein, Eric W. "Identity Matrix". mathworld.wolfram.com. Retrieved 2020-08-14.
  5. ^ Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of I2". The Mathematical Gazette 87, November 2003, 499–500.

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