Identity Matrix

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Identity Matrix

:

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 \
dots & dots & \ddots & dots \
0 & 0 & \cdots & 1 \end{bmatrix}

The important property of

I_n

is that
:

AI_n = A

and

I_nB = B

whenever these Matrix Multiplication s 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 Transformation s from an ''n''-dimensional vector space to itself, ''I_n'' represents the Identity Function , regardless of the Basis .

The ''i''th column of an identity matrix is the Unit Vector ''e_i''. The unit vectors are also the Eigenvector s 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}

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