Complex coordinates (z, w) on M_2(R): determinant and singular values

Idea. Every real-linear map of $\mathbb{C}$ splits uniquely into a complex-linear part $x\mapsto zx$ (a rotation-dilation) and a conjugate-linear part $x\mapsto w\bar x$ (a reflection-dilation). The coordinates $(z,w)$ make the geometry of $M_2(\mathbb{R})$ explicit.

Part 1. First,
$$z+w=\frac{2a+i(2c)}2=a+ic,\qquad z-w=\frac{2d+i(-2b)}2=d-ib.$$
With $x=x_1+ix_2$ we have $zx+w\bar x=(z+w)x_1+i(z-w)x_2=(a+ic)x_1+(id+b)x_2$, whose real and imaginary parts are $ax_1+bx_2$ and $cx_1+dx_2$. That is exactly $Ax$ read as a complex number.

(Conversely $a=\operatorname{Re}(z+w)$, $c=\operatorname{Im}(z+w)$, $d=\operatorname{Re}(z-w)$, $b=-\operatorname{Im}(z-w)$, so $A\mapsto(z,w)$ is a linear bijection $M_2(\mathbb{R})\to\mathbb{C}^2$.)

Part 2. Expanding,
$$|z|^2-|w|^2=\frac{(a+d)^2+(c-b)^2-(a-d)^2-(b+c)^2}4=\frac{4ad+(c-b)^2-(b+c)^2}4=\frac{4ad-4bc}4=ad-bc.$$

Part 3. For $x=e^{i\varphi}$ we have $|zx+w\bar x|^2=|z|^2+|w|^2+2\operatorname{Re}\bigl(zx\cdot\overline{w\bar x}\bigr)$ and $\overline{w\bar x}=\bar wx$, so
$$|Ax|^2=|z|^2+|w|^2+2\operatorname{Re}\bigl(z\bar w\,e^{2i\varphi}\bigr).$$
As $\varphi$ runs through $[0,2\pi)$, the number $z\bar we^{2i\varphi}$ runs around the circle of radius $|z||w|$, and its real part takes all values in $[-|z||w|,|z||w|]$. Hence $|Ax|^2$ ranges over
$$\bigl[|z|^2+|w|^2-2|z||w|,\ |z|^2+|w|^2+2|z||w|\bigr]=\bigl[(|z|-|w|)^2,(|z|+|w|)^2\bigr].$$
Now $|Ax|^2=x^TA^TAx$ on the unit circle ranges exactly between the smallest and the largest eigenvalue of $A^TA$, that is between $\sigma_2^2$ and $\sigma_1^2$. Therefore $\sigma_1=|z|+|w|$ and $\sigma_2=\bigl||z|-|w|\bigr|$.

Part 4. $\|A\|_F^2=\sigma_1^2+\sigma_2^2=(|z|+|w|)^2+(|z|-|w|)^2=2(|z|^2+|w|^2)$. (Directly: $|z|^2+|w|^2=\frac{(a+d)^2+(a-d)^2+(c-b)^2+(b+c)^2}4=\frac{a^2+b^2+c^2+d^2}2$.)

Check (numerical). For $200$ random $2\times2$ matrices, all four identities hold to machine precision. For the rotation $R(\theta)$ one finds $z=e^{i\theta}$, $w=0$, and for $\mathrm{diag}(1,-1)$ one finds $z=0$, $w=1$.

Remark. $\det$ is a quadratic form of signature $(2,2)$ on $M_2(\mathbb{R})\cong\mathbb{R}^4$: $|z|^2$ contributes two positive squares and $-|w|^2$ two negative ones. The formula $\|A\|=|z|+|w|$ for the operator norm shows that its unit ball is the $\ell^1$-sum of two Euclidean discs.

Lesson. In $M_2(\mathbb{R})$, choose coordinates adapted to the action of $SO(2)\times SO(2)$ (rotations on both sides): $z$ and $w$ are the natural ones.