Complex number
Matrix representation of complex numbers

Complex numbers a + bi can also be represented by 2 × 2 matrices that have the form (a − b b a). {\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.} Here the entries a and b are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of 2 × 2 matrices.

A simple computation shows that the map a + i b ↦ (a − b b a) {\displaystyle a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}} is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.

The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector (x, y) corresponds to the multiplication of x + iy by a + ib. In particular, if the determinant is 1, there is a real number t such that the matrix has the form

(cos t − sin t sin t cos t). {\displaystyle {\begin{pmatrix}\cos t&-\sin t\\\sin t&\;\;\cos t\end{pmatrix}}.} In this case, the action of the matrix on vectors and the multiplication by the complex number cos t + i sin t {\displaystyle \cos t+i\sin t} are both the rotation of the angle t.
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