Identity matrix

From Wikipedia, the free encyclopedia

In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and 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.)

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

Some mathematics books use U and E to represent the Identity Matrix (meaning "Unit Matrix" and "Elementary Matrix" respectively), although I is considered more universal.

The important property of $I n$ is that

A I n = A

and

I n B = B

whenever these matrix multiplications are defined. 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 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.

The ith column of an identity matrix is the unit vector e_i. 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: