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. It is denoted by In, 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.)

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_mA = AI_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. (The identity matrix itself is 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 ith 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 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:

I_n = \mathrm{diag}(1,1,...,1). \,

It can also be written using the Kronecker delta notation:

(I_n)_{ij} = \delta_{ij}. \,

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.

See also

Notes

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

External links