Editing Imaginary unit (section)

===Matrices===
Using the concepts of [[matrix (mathematics)|matrices]]  and [[matrix multiplication]], complex numbers can be represented in linear algebra. The real unit {{math|1}} and imaginary unit {{mvar|i}} can be represented by any pair of matrices {{mvar|I}} and {{mvar|J}} satisfying {{math|1=''I''{{isup|2}} = ''I'',}} {{math|1=''IJ'' = ''JI'' = ''J'',}} and {{math|1=''J''{{isup|2}} = −''I''.}} Then a complex number {{math|''a'' + ''bi''}} can be represented by the matrix {{math|''aI'' + ''bJ'',}} and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.

The most common choice is to represent {{math|1}} and {{mvar|i}} by the {{math|2&hairsp;×&hairsp;2}} [[identity matrix]] {{mvar|I}} and the matrix {{mvar|J}},

<math display=block>
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad
J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.</math>

Then an arbitrary complex number {{math|''a'' + ''bi''}} can be represented by:

<math display=block>aI + bJ = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.</math>

More generally, any real-valued {{math|2&hairsp;×&hairsp;2}} matrix with a [[trace (linear algebra)|trace]] of zero and a [[determinant]] of one squares to {{math|−''I''}}, so could be chosen for {{mvar|J}}. Larger matrices could also be used; for example, {{math|1}} could be represented by the {{math|4&hairsp;×&hairsp;4}} identity matrix and {{mvar|i}} could be represented by any of the [[Dirac matrices]] for spatial dimensions.