Vtyjkyuk

I am reading a paper, "A Morphable Model For The Synthesis Of 3D Faces" paper

I have a question about eq(3).

Solving a linear regression problem with data $D=(S_i, T_i)$ and target values $mu_i$, some vector can be calculated like eq(3)

$$
Delta S = sum_i=1^m mu_i (S_i - barS), qquad Delta T = sum_i=1^m mu_i (T_i - barT)
$$

In this paper eq(3) defines the direction with minimum mahalanobis distance to achieve a specific change $Delta mu$.

Author said that " It can be shown that Equation (3) defines the direction with minimal variance-normalized length $lVert Delta S rVert^2_M = leftlangle Delta S, C^-1_S Delta S rightrangle$, $lVert Delta T rVert^2_M = leftlangle Delta T, C^-1_T Delta T rightrangle$", but the proof is not shown.

Actually, I had solved following optimization problem on a regression problem using data some 2 dimensional data to verify numerically.

First of all, I had solved a regression problem on following data.
The first column is $mu$, the second and third are $S$, $T$ respectively.

$$
mathcalD=beginbmatrix
132 & 52 & 173 \
143 & 59 & 184 \
153 & 67 & 194 \
162 & 73 & 211 \
154 & 64 & 196 \
168 & 74 & 220 \
137 & 54 & 188 \
149 & 61 & 188 \
159 & 65 & 207 \
128 & 46 & 167 \
166 & 72 & 217
endbmatrix
$$

And optimization problem is formulated as follow

$$
beginaligned
&textmin. qquad p_11 Delta s^2 + 2 p_12 Delta s Delta t + p_22 Delta t^2 \[5pt]
&texts.t. qquad w_1 Delta s + w_2Delta t + w_0 = mu_0 + Delta mu - w_1 s_0 - w_2 t_0
endaligned
$$

where objective function is mahalanobis distance between $(s_0, t_0)$, and a point having objective function value $mu_0 + Delta mu$.
$mu_0$ is regression function value at $(s_0, t_0)$.
$p_11$, $p_12$, $p_22$ are components of inverse covariance matrix.
$w_0$, $w_1$, and $w_2$ are linear regression coefficients.

As a result of solving the above optimization problem, the $(Delta S, Delta T)=(1215.5455, 2290.)$ by eq(3) and the solution vector $(Delta s^*, Delta t^*)=(5.361, 10.0997)$ have the same direction.

How can this result be algebraically proven?