= Invariance of noise-whitened spectral angles to the whitening factor
{title2=$W_2=QW_1,\quad QQ^T=I$}
\b[Invariance of noise-whitened spectral angles to the whitening factor] means that any two complete whitening transformations for the same positive-definite <covariance> give the same angles between transformed shapes. Suppose square invertible matrices satisfy $W_iCW_i^T=I$. Put $Q=W_2W_1^{-1}$. Since $C=W_1^{-1}W_1^{-T}$,
$$
QQ^T=W_2CW_2^T=I,\qquad W_2=QW_1.
$$
Thus $Q$ is orthogonal. It may rotate or reflect the whitened coordinates, but preserves their inner products, norms and angles. The <noise-whitened angle between spectral signal and nuisance shapes> is consequently a property of the shapes and <covariance>, not a particular whitening algorithm.
For an original example, choose
$$
C=\begin{pmatrix}4&2\\2&2\end{pmatrix},\quad
W_1=\begin{pmatrix}1/2&0\\-1/2&1\end{pmatrix},\quad
Q=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.
$$
Direct multiplication gives $W_1CW_1^T=I$. With $\mathbf s=(2,1)^T$ and $\mathbf x=(2,2)^T$, the whitened shapes are $(1,0)^T$ and $(1,1)^T$, meeting at $45^{\circ}$. Using $W_2=QW_1$ instead gives $(0,1)^T$ and $(-1,1)^T$, again at $45^{\circ}$, although the coordinate components differ.
SciPy describes <covariance> whitening as transforming variables to unit <covariance> and illustrates it with a precision-matrix construction.<SciPy 1.16.2 documentation>[scipy.stats.Covariance.whiten, Description and Examples]{ref} The matrix proof above establishes the invariance independently of an implementation's chosen factor.
The same principle requires <covariance-consistent rescaling of whitened spectral coordinates>. If raw measurement units change by an invertible matrix $D$, transform both shapes by $D$ and <covariance> by $C'=DCD^T$; then $W'=W_1D^{-1}$ reproduces the same whitened shapes. Changing a plotted channel's scale without transforming its noise model is not this invariant operation.
The claim assumes a full whitening of a positive-definite <covariance>. Truncating modes can remove information and change angles, while a singular <covariance> requires restriction to its supported range. Uncertainty in an estimated <covariance> is also separate from this exact algebraic invariance.
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