# complex numbers and 2x2 matrices

Is it correct that set ${\mathbb C}$ is isomorphic to the set of following 2x2 matrices: $$\left( \begin{array}{cc} a &-b\\ b &a \end{array}\right)$$ $a \in {\mathbb R}$ and $b \in {\mathbb R}$?

In other words: are these two sets identical?

• Yes, these are isomorphic fields, but no, that does not mean the two sets are equal. – Ittay Weiss Nov 17 '13 at 18:27
• Actually a meant that they are "identical" (with quotes). Sometimes when there is isomorphism between two sets it is said that they are alike. – mechanician Nov 17 '13 at 18:29
• the term is 'essentially the same'. – Ittay Weiss Nov 17 '13 at 18:30

Yes. Provided that $a^2+b^2 \neq 0$, that means the determinant is non-zero.
the map $\varphi: (\mathbb{C}-\{0\},.) \to (M, \times)$ given by $(a+bi) \mapsto \left( \begin{array}{cc} a &-b\\ b &a \end{array}\right)$ is an isomorphism. You can check it easily ($M$ is the set of all such matrices and $\times$ is matrix multiplication)
• What is the reason for $a^2 + b^2 \neq 0$? I think that zero matrix is absolutely equivalent to zero complex number, isn't it? – mechanician Nov 17 '13 at 18:12
• Should we not have $a^2+b^2 \ge 0$, because the absolute value of a complex number cannot be smaller than zero? – asmaier Apr 10 '14 at 20:14