C. Congruence transformations

The congruence transformation is defined in Eqs. (3) and (14). Substituting the polar decomposition for $\vec{X}$ we get Eq. (15):

$$\begin{eqnarray*}\vec{B}' = xx^* \,\vec{H}\ (\vec{Y}\vec{B}\vec{Y}^{\dagger})\ \vec{H}^{\dagger}. \end{eqnarray*}$$

Since $\vec{H}$ and $\vec{Y}$ are unimodular, $\det \vec{B}' = (xx^*)^2 \det \vec{B}$ or, apart from the scale factor

$$\begin{eqnarray*}b'^2 -\vec{b}'^2 = b^2 - \vec{b}^2 . \end{eqnarray*}$$

(In the main text of this paper, b and $\vec{b}$ are written as I and $\vec{p}$, respectively, to emphasize the physical interpretation of $\vec{B}$ as a brightness.)

The effect of the component unitary and positive hermitian transformations can now be analysed by replacing $\vec{B}$, $\vec{Y}$ and $\vec{H}$ with their equivalent quaternions and carrying out the multiplications:

  
C.1. Unitary transformations

Consider the most general unitary transformation

$$\begin{eqnarray*}\lefteqn{ [\, b'+\vec{b}'_{\scriptscriptstyle\parallel \vphanto... ...\rm e}^{-i\psi} [\, \cos\eta {-} i\vec{1}_{\vec{y}}\sin\eta \,] \end{eqnarray*}$$

where $\vec{b}_{\scriptscriptstyle\parallel \vphantom{'}}$ and $\vec{b}_{\scriptscriptstyle\perp \vphantom{'}}$ are the components of $\vec{b}$ that are parallel and perpendicular to $\vec{1}_{\vec{y}}$.

The unitary quaternions are collinear with $[\, b + \vec{b}_{\scriptscriptstyle\parallel \vphantom{'}}\,]$ so

$$\begin{eqnarray*}[\, b'+\vec{b}'_{\scriptscriptstyle\parallel \vphantom{'}}\,]= [\, b+\vec{b}_{\scriptscriptstyle\parallel \vphantom{'}}\,] . \end{eqnarray*}$$

For $\vec{b}_{\scriptscriptstyle\perp \vphantom{'}}$ we must carry out the multiplications, and obtain

$$\begin{eqnarray*}[\, \vec{b}'_{\scriptscriptstyle\perp \vphantom{'}}\,] = [\, \... ...vec{b}_{\scriptscriptstyle\perp \vphantom{'}}\,\sin 2\eta \,] . \end{eqnarray*}$$

That is, $\vec{b}'_{\scriptscriptstyle\perp \vphantom{'}}$ is a copy of $\vec{b}_{\scriptscriptstyle\perp \vphantom{'}}$ rotated over an angle $2\eta$. The two results combined show that a unitary transformation $\vec{Y}$ leaves the scalar part of its input invariant and rotates its vector part around the axis $\vec{1}_{\vec{y}}$ over an angle $2\eta$; this is the polrotation effect of Sect. 5.1. The scalar and vector parts are transformed independently.

The vector $\vec{1}_{\vec{y}} \sin\eta$ that characterises the rotation is known as the Gibbs vector ([Korn & Korn 1961]). Vectors collinear with it are invariant; they are eigenvectors of the rotation.

From the theory of linear transformations I take the result that the mathematical expression for a rotation is

$$\begin{eqnarray*}\vec{b}' = \vec{R}\,\vec{b} \end{eqnarray*}$$

where the $3 \times 3$ transformation matrix $\vec{R}$ is real, orthogonal and unitary. Its unitarity guarantees the invariance of scalar products and in particular of real vector lengths.

The Euclidian rotations of the vector part $\vec{b}$ form a subset of the general pseudo-Euclidian rotations represented by the Lorentz transformation ([Feynman et al. 1975]), just as the unitary transformations form a subset of the general congruence transformations.

  
C.2. The fundamental unitary matrices

Any rotation in a three-dimensional Euclidian space can be represented as a succession of three rotations around mutually perpendicular axes ([Korn & Korn 1961]). Choosing for these axes the three base vectors of Eq. (28) we find the three respective basic unimodular unitary matrices

• The phase-difference transformation

 

$$\vec{Y}_{\mathbf{q}}(\phi) = \left( \begin{array}{cc} \exp i\phi & 0 \\ 0 &\exp -i\phi \end{array} \right) = \exp i\,[\vec{1}_{\mathbf{q}}]\phi .$$ (36)

• The ellipticity transformation

 

$$\vec{Y}_{\mathbf{u}}(\epsilon) = \left( \begin{array}{rr} \cos\ep... ...n & \cos\epsilon \end{array} \right) = \exp i\,[\vec{1}_{\mathbf{u}}]\epsilon .$$ (37)

• The (feed or xy frame) rotation transformation

 

$$\vec{Y}_{\mathbf{v}}(\theta) = \left( \begin{array}{rr} \cos\thet... ...\theta &\cos\theta \end{array} \right) = \exp i\,[\vec{1}_{\mathbf{v}}]\theta .$$ (38)

  
C.3. The polrotation in circular coordinates

It is of some interest to consider the form that Eq. (18) takes in circular coordinates. The feed and receiver terms $\vec{Y}_{\mathbf{u}}$ and $\vec{Y}_{\mathbf{q}}$ operate on the signal vector formed by the l and r voltages in the feed-receiver system and need not change. The geometric rotation term $\vec{Y}_{\mathbf{v}}(\theta)$ must be transformed to the circular lr coordinate frame (Paper I) in which the radiation is measured. From Paper I we take the result that this transformation transforms Stokes V into Q, hence $\vec{1}_{\mathbf{v}}$ into $\vec{1}_{\mathbf{q}}$. Thus $\vec{Y}_{\mathbf{v}}(\theta)$ assumes the form $\vec{Y}_{\mathbf{q}}(\theta)$, and the equivalent of Eq. (18) becomes

 

$$\vec{Y}= \vec{Y}_{\mathbf{q}}(\phi) \,\vec{Y}_{\mathbf{u}}(\epsilon) \,\vec{Y}_{\mathbf{q}}(\theta) .$$ (39)

This is another generic way of factoring an arbitrary rotation ([Korn & Korn 1961]).

When the feed-error term $\vec{Y}_{\mathbf{u}}$ is reduced to unity, the remaining two $\vec{Y}_{\mathbf{q}}$ terms fuse into one. The usual practice of merging them regardless of the feed errors amounts to inverting the order of the factors in Eq. (39); this is justifiable only in the quasi-scalar approximation (cf. Sect. 7).

  
C.4. Positive hermitian transformations

The treatment of the unimodular positive hermitian transformation is analogous to that of the unitary one. Starting from Eq. (33) for $\vec{H}$ one finds

 

$$\begin{array}{lclrrlr} b' &=& b& &\cosh 2\gamma \ + & \vec{b}_{\s... ...\multicolumn{2}{l}{\vec{b}_{\scriptscriptstyle\perp \vphantom{'}}.} \end{array}$$ (40)

The effect is in a sense complementary to that of the unitary transformation, but there is no analogous geometric interpretation. The scalar and vector parts are not transformed independently but get mixed: this is the polconversion effect of Sect. 5.1.

There is no interaction between vector components in different directions: The transformation is said to be rotation-free.

  
C.5. Minimal-variance theorem

Of all complex numbers $z = a\,{\rm e}^{i\phi}$, z=a has the smallest distance squared |(z-1)|² to unity. I shall now prove an analogous property for $2 \times 2\$matrices:

Theorem: Let $\vec{H}$ be a given positive hermitian and $\vec{Y}$ an arbitrary unitary matrix. Of all products $\vec{X}= \vec{Y}\vec{H}$, $\vec{X}=\vec{H}$ minimises the quantity

$$\begin{eqnarray*}F_{\vec{H}\vphantom{'}}(\vec{Y}) = \mbox{Var}\ (\vec{Y}\vec{H}- \mathbf{I}). \end{eqnarray*}$$

Using the definition of $\mbox{Var}\$, the commutation and transposition invariance of $\mbox{Tr}\$ and the hermiticity of $\vec{H}$, convert the equation to

 

$$\begin{array}{rcl} F_{\vec{H}\vphantom{'}}(\vec{Y}) &=& \mbox{Tr}... ...mbox{Tr}\ (\vec{H}\vec{H}+ \mathbf{I}- 2{\rm Re\,}\vec{Y}\vec{H}) . \end{array}$$ (41)

To find the minimum, consider the term

 

$$\begin{array}{rcl} \mbox{Tr\,Re\ }\vec{Y}\vec{H} &=& \mbox{Tr\,Re... ...psi \cos\eta -\vec{1}_{\vec{y}} \cdot \vec{h}\, \sin\psi \sin\eta . \end{array}$$ (42)

Differentiation with respect to $\psi$ and $\eta$ gives the equations

$$\begin{eqnarray*}h \,\sin\psi \,\cos\eta + \vec{1}_{\vec{y}} \cdot \vec{h}\,\cos... ...n\eta + \vec{1}_{\vec{y}} \cdot \vec{h}\,\sin\psi \,\cos\eta =0 \end{eqnarray*}$$

whose obvious solutions are $\cos\psi = \cos\eta =0$ and $\sin\psi = \sin\eta = 0$.

Since $\vec{H}$ is positive hermitian,

$$\begin{eqnarray*}h > \vert\vec{h}\vert \geq \vec{1}_{\vec{y}} \cdot \vec{h} \end{eqnarray*}$$

so a. there are no other solutions and b. the former one, which is equivalent to $\vec{Y}=\mathbf{I}$, maximises Eq. (42) and hence minises $F_{\vec{H}}(\vec{Y})$. Moreover this is true for all directions $\vec{1}_{\vec{y}}$.