Every matrix in $SU(2)$ can be written as: $P= I\cos \theta+ A\sin \theta$, $A$ on the equator.

Question

How can I show that every matrix in $SU(2)$ can be written as: $P=I\cos \theta + A\sin \theta$, with $A$ on the equator?

Answer 1

Hint:

Write down a general matrix

$$P = \left(\matrix{a& b\\c & d}\right)$$

and enforce $PP^\dagger =I$ and $\det P = 1$ which gives you three independent equations relating $a,b,c,d$ and use this to show that a general matrix in $SU(2)$ can be written (NB: not the same $a,b$ as above):

$$P = \left(\matrix{a& -b^*\\b^* & a^*}\right)$$

with $|a|^2 + |b|^2 = 1$ and $^*$ denotes complex conjugation. Now rewrite $P$ as

$$P = \left(\matrix{\Re a& 0\\0 & \Re a}\right) + \left(\matrix{\Im a& -b^*\\b^* & -\Im a}\right)$$

where $\Re a$ is the real part of $a$ and $\Im a$ is the imaginary part of $a$.