The identity matrix is a the simplest nontrivial diagonal matrix, defined such that
 |
(1)
|
for all vectors 
. An identity matrix may be denoted 
, 
,
(the latter being an abbreviation for
the German term "Einheitsmatrix"; Courant and Hilbert 1989, p. 7),
or occasionally 
,
with a subscript sometimes used to indicate the dimension of the matrix. Identity
matrices are sometimes also known as unit matrices (Akivis and Goldberg 1972, p. 71).
The 
identity matrix is given explicitly by
 |
(2)
|
for 
,
..., 
,
where 
is the Kronecker delta. Written explicitly,
![I=[1 0 ... 0; 0 1 ... 0; | | ... |; 0 0 ... 1].](/images/equations/IdentityMatrix/NumberedEquation3.svg) |
(3)
|
The 
identity matrix is implemented in the Wolfram
Language as IdentityMatrix[n].
"Square root of identity" matrices can be defined for 
by solving
![[a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); | ... ... |; a_(n1) a_(n2) ... a_(nn)][a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); | ... ... |; a_(n1) a_(n2) ... a_(nn)]=[1 0 ... 0; 0 1 ... 0; | | ... 0; 0 0 ... 1].](/images/equations/IdentityMatrix/NumberedEquation4.svg) |
(4)
|
For 
,
the most general form of the resulting square root matrix is
![I_2^(1/2)=[+/-d (1-d^2)/c; c ∓d],[+/-d c; (1-d^2)/c ∓d]](/images/equations/IdentityMatrix/NumberedEquation5.svg) |
(5)
|
giving
![[+/-1 0; 0 +/-1],[+/-1 0; c ∓1],[+/-1 c; 0 ∓1]](/images/equations/IdentityMatrix/NumberedEquation6.svg) |
(6)
|
as limiting cases.
"Cube root of identity" matrices can take on even more complicated forms. However, one simple class of such matrices is called k-matrices.
