 Invariance of noise-whitened spectral angles to the whitening factor

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 $W_{i}C{W}_i^T=I$. Put $Q=W_2{W}_1^{-1}$. Since $C=W_1^{-1}{W}_1^{-T}$,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, chooseDirect multiplication gives $W_{1}C{W}_1^T=I$. 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 $W_2=Q{W}_1$ 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.^[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 $C^{\prime }=DC{D}^T$; then $W^{\prime }=W_1{D}^{-1}$ 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.