- Identity matrix
linear algebra, the identity matrix or unit matrix of size "n" is the "n"-by-"n" square matrixwith ones on the main diagonaland zeros elsewhere. It is denoted by "I""n", or simply by "I" if the size is immaterial or can be trivially determined by the context. (In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to "I".)
Some mathematics books use "U" and "E" to represent the Identity Matrix (meaning "Unit Matrix" and "Elementary Matrix", or from the German "Einheitsmatrix" ["Identity Matrix"; On Wolfram's MathWorld; http://mathworld.wolfram.com/IdentityMatrix.html] , respectively), although "I" is considered more universal.
The important property of
matrix multiplicationof identity matrix is that for "m"-by-"n" "A":In particular, the identity matrix serves as the unit of the ring of all "n"-by-"n" matrices, and as the identity elementof the general linear groupGL("n") consisting of all invertible "n"-by-"n" matrices. (The identity matrix itself is obviously invertible, being its own inverse.)
Where "n"-by-"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 "i"th column of an identity matrix is the
unit vector"ei". The unit vectors are also the eigenvectors of the identity matrix, all corresponding to the eigenvalue 1, which is therefore the only eigenvalue and has multiplicity"n". It follows that the determinantof 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::
It can also be written using the
The identity matrix also has the property that, when it is the product of two square matrices, the matrices can be said to be the inverse of one another.
*planetmath reference|title=Identity matrix|id=1223
Wikimedia Foundation. 2010.
Look at other dictionaries:
identity matrix — vienetinė matrica statusas T sritis fizika atitikmenys: angl. identity matrix; unit matrix vok. Einheitsmatrix, f rus. единичная матрица, f pranc. matrice unité, f; matrice unitaire, f … Fizikos terminų žodynas
identity matrix — noun a scalar matrix in which all of the diagonal elements are unity • Syn: ↑unit matrix • Hypernyms: ↑scalar matrix … Useful english dictionary
identity matrix — noun A diagonal matrix all of the diagonal elements of which are equal to 1. Syn: unit matrix … Wiktionary
identity matrix — noun Date: circa 1929 a square matrix that has numeral 1 s along the principal diagonal and 0 s elsewhere … New Collegiate Dictionary
identity matrix — Math. a matrix that has 1 in each position on the main diagonal and 0 in all other positions. [1940 45] * * * … Universalium
Identity — may refer to:Philosophy* Identity (philosophy), the sameness of two things * Identity theory of mind, in the philosophy of mind, holds that the mind is identical to the brain * Personal identity (philosophy) * Identity (social science) * Identity … Wikipedia
Matrix normal distribution — parameters: mean row covariance column covariance. Parameters are matrices (all of them). support: is a matrix … Wikipedia
Matrix multiplication — In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n by m matrix and B is an m by p matrix, the result AB of their multiplication is an n by p matrix defined only if… … Wikipedia
Matrix exponential — In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.… … Wikipedia
Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… … Wikipedia