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Covariance-consistent rescaling of whitened spectral coordinates changes measurement units or mixes coordinates while transforming their noise model at the same time. If an invertible matrix $D$ maps a spectral vector to ${\mathbf{y}}^{\prime }=D\mathbf{y}$, thenThe covariance follows directly from the expectation of the transformed error outer product. For positive-definite $C$,Thus the inverse-covariance inner product, its norms and the resulting whitened angle are invariant. Invariance of noise-whitened spectral angles to the whitening factor establishes the related freedom to rotate the fully whitened coordinates.

For an original example, let $C=\mathrm{diag}(1,9)$, $\mathbf{s}=(1,1{)}^T$ and $\mathbf{x}=(1,-1{)}^T$. Their noise-whitened angle has cosine $0.8$, or angle about $36.87^{\circ }$. Converting only the first raw coordinate's numerical units by a factor of 1000 gives $D=\mathrm{diag}(1000,1)$ and $C^{\prime }=\mathrm{diag}(10^6,9)$. Using that transformed covariance keeps the cosine exactly $0.8$.

If the shapes are rescaled but the old covariance is mistakenly retained, the calculated cosine becomes approximately $0.99999978$, suggesting an angle only about $0.0382^{\circ }$. A unit conversion would then appear to destroy the separation between signal and nuisance. That apparent change comes from inconsistent uncertainty accounting.

For any full whitening factor $W$, the transformed factor $W^{\prime }=W{D}^{-1}$ satisfies $W^{\prime }{C}^{\prime }(W^{\prime }{)}^T=I$ and gives $W^{\prime }{\mathbf{s}}^{\prime }=W\mathbf{s}$. The pinned SciPy documentation describes whitening and demonstrates a near-identity covariance after transformation.^[ref] Measurement propagation likewise retains all covariance terms under a linear transformation.^[ref]

The invertibility condition matters. Noninvertible spectral compression versus a coordinate change can discard information and need not preserve these angles. A transformation estimated from the same noisy observations also brings estimation uncertainty beyond the fixed-matrix identity proved here.

The noise-whitened angle between spectral signal and nuisance shapes measures their separation in units of measurement noise rather than raw plotting units. For nonzero shapes $\mathbf{s},\mathbf{x}$ and a positive-definite covariance $C$, choose a square whitening transformation $W$ with $WC{W}^T=I$. The angle isIt is the ordinary Euclidean angle after whitening. Both parallel and antiparallel shapes are indistinguishable up to an unknown nuisance amplitude; the precision penalty depends on ${\sin }^2\varphi$, as in noise penalty from nearly parallel signal and reference shapes.

For an original example, take raw shapes $\mathbf{s}=(1,1{)}^T$, $\mathbf{x}=(1,-1{)}^T$ and $C=\mathrm{diag}(1,100)$. The plotted shapes are orthogonal, but whitening gives $(1,0.1{)}^T$ and $(1,-0.1{)}^T$. Thus $\cos \varphi =0.99/1.01\approx 0.98020$ and $\varphi \approx 11.42^{\circ }$. The second channel, which separates their signs, is noisy. Rejecting the nuisance multiplies the optimum variance by $1/{\sin }^2\varphi =25.5025$, or standard uncertainty by $5.05$.

The invariance of noise-whitened spectral angles to the whitening factor follows from $W^{T}W=C^{-1}$. Different valid square whitening factors can rotate the whitened coordinates without changing the angle. Likewise an invertible change of measurement units transforms both shapes and covariance, leaving the inverse-covariance expression unchanged. Rescaling only the displayed curves while leaving their noise model fixed is a different operation.

The covariance must describe the actual measured channels and shared errors, using the full propagation rule.^[ref] If $C$ is singular, first restrict to its physically allowed range; an unrestricted inverse whitening factor does not exist.

The primary stimulated-Raman experiment described systematic spectral noise and subtraction of a pure-solvent reference.^[ref] The angle above assesses which desired patterns remain distinguishable after such nuisance rejection under a stated covariance. It does not establish that an estimated reference shape or noise model is itself accurate.