Square matrices and transformations of the plane

Every strictly positive real number has two square roots, as does every nonzero complex number. More generally, there are at most two square roots in any set known as a field. The rational numbers—that is, fractions—provide an example: for r₁ = 4/9, two numbers have r₁ as their square, namely 2/3 and ‒2/3; r₂ = 17 has no square root in this set, since the real number $\sqrt{17}\backslash sqrt\{17\}$ is irrational, so r₂ is not the square of any fraction. In some more general sets, however, the number of square roots may far exceed two.

A little vector geometry… -------------------------

The set of square matrices of a given size is equipped with addition and multiplication (see box).

In a Euclidean vector plane P with an orthonormal basis $(\overrightarrow{i},\overrightarrow{j}),\backslash left(\; \backslash overrightarrow\; \{i\},\; \backslash overrightarrow\{j\}\; \backslash right),$ the Euclidean norm of a vector $\overrightarrow{u}\backslash overrightarrow\; \{u\}$ with coordinates (x, y) in this basis is $\mathrm{\parallel }\overrightarrow{u}\mathrm{\parallel }=\mathrm{\parallel }(x,y)\mathrm{\parallel }=\sqrt{x^2+y^2}\mathrm{.}\backslash Vert\; \backslash overrightarrow\; \{u\}\; \backslash Vert\; =\; \backslash Vert\; (x,\; y)\; \backslash Vert\; =\; \backslash sqrt\{x^2+y^2\}.$