Identity matrix

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. :

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}

The important property of ''I_n'' is that :''AI_n'' = ''A'' and ''I_nB'' = ''B'' whenever these matrix multiplications are defined. In particular, the identity matrix serves as the unit of the ring (mathematics) of all ''n''-by-''n'' matrices, and as the identity element of the general linear group GL(''n'') consisting of all invertible matrix ''n''-by-''n'' matrices. (The identity matrix itself is obviously invertible, being its own inverse.) The ''i''th column of an identity matrix is the unit vector ''e_i''. Using the notation that is sometimes used to concisely describe diagonal matrix, 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}

Abstract algebra Linear algebra Matrices

See other meanings of words starting from letter:

I

IA | IB | IC | ID | IE | IF | IG | IH | IJ | IK | IL | IM | IN | IO | IP | IR | IS | IT | IU | IW | IX | IY | IZ |

Words begining with Identity_matrix:

Identity_matrix