Vtgyjfy

I've got a problem that I'm struggling to put into a form that I can analyze.

Suppose I have a quadratic form $f(x,y)=ax^2+2bxy+cy^2 = mathbf{u}mathbf{A}mathbf{u}^T$ for $mathbf{u} = begin{bmatrix}x&yend{bmatrix}$, $mathbf{A} = begin{bmatrix}a & b cr b &cend{bmatrix}$

Now I have two series of coordinate pairs $mathbf{u}_k = begin{bmatrix}x_k&y_kend{bmatrix}$ and $mathbf{v}_k = begin{bmatrix}x_k&y_kend{bmatrix}$, and I want to compare the sets $U={mathbf{u}_k}$ and $V={mathbf{v}_k}$ by evaluating the metrics

$$g(U) = maxlimits_{lVert{A}rVert le 1} sumlimits_k f(x_k,y_k)$$

and comparing $g(U)$ and $g(V)$.

Is there an easy way to compute $g(U)$ from knowing the individual $(x_k,y_k)$ pairs? I can compute the following quantities easily:

$$begin{align}
S_{xx} &= sum x^2 cr
S_{xy} &= sum xy cr
S_{yy} &= sum y^2
end{align}$$

I just am not sure how to use those to compute the maximum over the constraint $lVert{A}rVert le 1$. (I have a very poor knowledge of matrix norm identities.)

(Important fact: I'm using Frobenius norms.)

Given the Frobenius norm definition, this means that I'm looking for the maximum value of $aS_{xx} + 2bS_{xy} + cS_{yy}$ subject to the constraint $a^2 + 2b^2 + c^2 le 1$.

Not sure where to go from here, though it seems like I'm really close.

Using method of Lagrange multipliers you can find that

$$
underset{a^2 + 2b^2 + c^2 le 1}{operatorname{max}} Big(aS_{xx} + 2bS_{xy} + cS_{yy}Big)=sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2},
$$

which is obtained when

$$
a=frac{S_{xx}}{sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}}, quad b=frac{S_{xy}}{sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}}, quad c=frac{S_{yy}}{sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}}
$$

Hmmm. If I write $u=a, v=2b, w=c$ then I have a 3-dimensional vector $mathbf{r} = (u,v,w)$ of magnitude 1 or less, and the function $g(S_{xx},S_{xy},S_{yy}) = uS_{xx}+vS_{xy}+wS_{yy} = mathbf{r}cdotmathbf{S}$ for $mathbf{S} = (S_{xx},S_{xy},S_{yy})$ is maximized when $mathbf{r}$ and $mathbf{S}$ are in the same direction, or in other words, $mathbf{r} = frac{mathbf{S}}{lvertmathbf{S}rvert}$, in which case $g(mathbf{S}) = mathbf{r}cdotmathbf{S} = |mathbf{S}| = sqrt{S_{xx}{}^2+S_{xy}{}^2+S_{yy}{}^2}$.

Did I get that right?

Take 2:

If I write $u=a, v=sqrt{2}b, w=c$ then I have a 3-dimensional vector $mathbf{r} = (u,v,w)$ of magnitude 1 or less, and the function $g(S_{xx},S_{xy},S_{yy}) = uS_{xx}+sqrt{2}vS_{xy}+wS_{yy} = mathbf{r}cdotmathbf{S}$ for $mathbf{S} = (S_{xx},sqrt{2}S_{xy},S_{yy})$ is maximized when $mathbf{r}$ and $mathbf{S}$ are in the same direction, or in other words, $mathbf{r} = frac{mathbf{S}}{lvertmathbf{S}rvert}$, in which case $g(mathbf{S}) = mathbf{r}cdotmathbf{S} = |mathbf{S}| = sqrt{S_{xx}{}^2+2S_{xy}{}^2+S_{yy}{}^2}$.