For every real number a, there is a 2×2 diagonal matrix
|
é ë |
a 0 0 a |
ù û |
, |
each of whose diagonal entries is equal to a. Conversely, every such matrix determines a real number, namely the unique real number found on its diagonal. The matrices behave like the numbers, in the sense that the matrix corresponding to the sum of two numbers is the sum of the matrices corresponding to the numbers, and likewise for multiplication.
We can define a set C, consisting of the 2×2 matrices of the form
|
é ë |
a -b b a |
ù û |
. |
Then this set C is closed under the operations of addition, multiplication, transposition and inversion. With these operations, C becomes the field of complex numbers.
Usually the typical complex number is denoted, not by a square matrix, but by the linear polynomial
a + ib ,
where i is the unknown, but
i2 = -1 .
Then the operation of complex conjugation corresponds to taking the transpose of the matrix. Also, the modulus or absolute value of a complex number is the square root of the determinant of the corresponding matrix.
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