- Identity matrix
In

linear algebra , the**identity matrix**or**unit matrix**of size "n" is the "n"-by-"n"square matrix with ones on themain 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 asquantum mechanics , the identity matrix is denoted by a boldface one,**1**; otherwise it is identical to "I".):$I\_1\; =\; egin\{bmatrix\}1\; end\{bmatrix\},\; I\_2\; =\; egin\{bmatrix\}1\; 0\; \backslash 0\; 1\; end\{bmatrix\},\; I\_3\; =\; egin\{bmatrix\}1\; 0\; 0\; \backslash 0\; 1\; 0\; \backslash 0\; 0\; 1\; end\{bmatrix\},\; cdots\; ,\; I\_n\; =\; egin\{bmatrix\}1\; 0\; cdots\; 0\; \backslash 0\; 1\; cdots\; 0\; \backslash vdots\; vdots\; ddots\; vdots\; \backslash 0\; 0\; cdots\; 1\; end\{bmatrix\}$

Some mathematics books use "U" and "E" to represent the Identity Matrix (meaning "Unit Matrix" and "Elementary Matrix", or from the German "Einheitsmatrix" [

*"Identity Matrix"; On Wolfram's MathWorld; http://mathworld.wolfram.com/IdentityMatrix.html*] , 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 theidentity element of thegeneral 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 theidentity function , regardless of the basis.The "i"th column of an identity matrix is the

unit vector "e_{i}". The unit vectors are also theeigenvector s of the identity matrix, all corresponding to the eigenvalue 1, which is therefore the only eigenvalue and hasmultiplicity "n". It follows that thedeterminant 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.

**References****External links***planetmath reference|title=Identity matrix|id=1223

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