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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.

The noise penalty from nearly parallel signal and reference shapes is the precision lost when a spectral estimator must reject a nuisance pattern resembling its desired signal. For nonzero shapes $\mathbf{s},\mathbf{x}$ and positive-definite noise covariance $C$, define their inverse-covariance correlationThe Cauchy-Schwarz inequality gives $|r|\le 1$. The minimum-variance spectral contrast orthogonal to a reference shape has varianceIf the nuisance amplitude were known and subtracted without uncertainty, the corresponding optimum variance would be $V_{\mathrm{known}}=1/({\mathbf{s}}^T{C}^{-1}\mathbf{s})$. Exact nuisance rejection therefore multiplies variance by $1/(1-r^2)$ and standard uncertainty by $1/\sqrt{1-r^2}$.

The relevant angle is the noise-whitened angle between spectral signal and nuisance shapes: it is measured after a whitening transformation, not generally from the raw plotted spectra. A channel with large noise contributes little to separating the shapes even if their plotted curves differ there.

For an original example, let $C={\sigma }^{2}I$, $\mathbf{x}=(1,0{)}^T$ and $\mathbf{s}=(1,0.1{)}^T$. Rejecting the reference forces the contrast to $(0,10{)}^T$, with variance $100{\sigma }^2$. With known nuisance amplitude, its variance would be ${\sigma }^2/1.01$. The rejection penalty is therefore $101$ in variance, about $10.05$ in standard uncertainty. Only the weak second-channel component distinguishes signal from nuisance.

The primary stimulated-Raman experiment reported systematic spectral noise above its calculated shot-noise limit and described pure-solvent subtraction of a common pattern.^[ref] A reference can remove that shared pattern while also making an overlapping desired signal hard to estimate. Retain full covariance in the variance calculation.^[ref] At exact proportionality, no unit-response rejecting contrast exists; a low-noise result obtained by suppressing the signal as well is not a valid amplitude estimate.