Norm of a set of vectors with respect to a quadratic form

Question

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 $\mathbf{u} = \begin{bmatrix}x&y\end{bmatrix}$, $\mathbf{A} = \begin{bmatrix}a & b \cr b &c\end{bmatrix}$

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

$$g(U) = \max\limits_{\lVert{A}\rVert \le 1} \sum\limits_k f(x_k,y_k)$$

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.)

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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.

Answer 1

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 $g(S_{xx},S_{xy},S_{yy}) = uS_{xx}+vS_{xy}+wS_{yy} = \mathbf{r}\cdot\mathbf{S}$ 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, $\mathbf{r} = \frac{\mathbf{S}}{\lvert\mathbf{S}\rvert}$, in which case $g(\mathbf{S}) = \mathbf{r}\cdot\mathbf{S} = |\mathbf{S}| = \sqrt{S_{xx}{}^2+S_{xy}{}^2+S_{yy}{}^2}$.

Did I get that right?

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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 $g(S_{xx},S_{xy},S_{yy}) = uS_{xx}+\sqrt{2}vS_{xy}+wS_{yy} = \mathbf{r}\cdot\mathbf{S}$ 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, $\mathbf{r} = \frac{\mathbf{S}}{\lvert\mathbf{S}\rvert}$, in which case $g(\mathbf{S}) = \mathbf{r}\cdot\mathbf{S} = |\mathbf{S}| = \sqrt{S_{xx}{}^2+2S_{xy}{}^2+S_{yy}{}^2}$.

Answer 2

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

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