Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. Linear algebra is the branch of Mathematics concerned with In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, the main diagonal (sometimes leading diagonal or primary diagonal) of a matrix A is the collection of cells A_{ij} 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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons )

$I_1 = \begin{bmatrix} 1 \end{bmatrix} ,\ I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ,\ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ,\ \cdots ,\ I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}$

Some mathematics books use U and E to represent the Identity Matrix (meaning "Unit Matrix" and "Elementary Matrix", or from the German "Einheitsmatrix"^[1], respectively), although I is considered more universal.

The important property of matrix multiplication of identity matrix is that for m-by-n A

I m A = A I n = A

In particular, the identity matrix serves as the unit of the ring of all n-by-n matrices, and as the identity element of the general linear group GL(n) consisting of all invertible n-by-n matrices. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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 In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- (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, I_n represents the identity function, regardless of the basis. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

The ith column of an identity matrix is the unit vector e_i. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length 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. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes It follows that the determinant of the identity matrix is 1 and the trace is n. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal

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

I n = diag(1,1,. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero . . ,1).

It can also be written using the Kronecker delta notation:

(I n) i j = δ i j . In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

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.

References

^ "Identity Matrix"; On Wolfram's MathWorld; http://mathworld.wolfram.com/IdentityMatrix.html

External links

Identity matrix on PlanetMath

PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(linear algebra) A diagonal matrix all of the diagonal elements of which are equal to 1.

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