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I have problems with a generalized quadratic form defined on a non-default, multi dimensional inner product and the meaning of its eigenvalues.

Consider the following quadratic form $A(x)$

$A(x_1,x_2) = \alpha \|x_1\|^2 - 2(1-\alpha) \langle x_1,x_2 \rangle + \alpha \|x_2\|^2$

where $\|\cdot\|$ is a norm deduced from some arbitrary inner product $\langle \cdot, \cdot \rangle: H \times H \mapsto \mathbb{R}$ ($\|x_i\|^2 = \langle x_i,x_i \rangle$) with a Hilbert space $H$.

In the source* it is said that there exist the two eigenvalues $1$ and $2 \alpha - 1$ of $A$ and that $A$ is strictly elliptic if and only if $\alpha > 1/2$.

Usually in the literature quadratic forms are written as matrix vector multiplication or standard scalar products in $\mathbb{R}^n$, i.e. $x^T B x = x \cdot (Bx) = \sum_{i,j=1}^2 b_{ij} x_i x_j$

Therefore I'm looking for another inner product, we call it $[\cdot,\cdot]: H^2 \times H^2 \mapsto \mathbb{R}$, such that one can write $A(x) = [x, Bx] = [(x_1, x_2)^T, B(x_1,x_2)^T]$

These are my own thoughts so far:

Obviously $A$ can also be denoted as $A(x) = \langle x_1, \alpha x_1 \rangle + 2 \langle x_1, (1-\alpha) x_2 \rangle + \langle x_2, \alpha x_2 \rangle$

I define $[\cdot,\cdot]$ as a sum of the one-dimensional inner products $<\cdot,\cdot>$ on $H$: $[x,y] = [(x_1, x_2)^T , (y_1, y_2)^T] := \langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle$

Then there holds $A(x) = [x,Bx]$ with $B = \begin{bmatrix} \alpha & \alpha-1 \\ \alpha-1 & \alpha \end{bmatrix}$. The eigenvalues of $B$ are then equal to the eigenvalues of $A$ mentioned at the beginning.

So returning to my questions:

  • What is the geometrical meaning of these eigenvalues? $A$ maps to a scalar and not to a vector.

  • Is there any generalized definition of quadratic forms on arbitrary inner products or theoreom about representation as an arbitrary inner product? The previous derivation of this kind of quadratic form is just my "invention".

Thanks!


The source: Jérôme Jaffré, Vincent Martin, Jean Roberts. Modeling Fractures and Barriers as Interfaces for Flow in Porous Media. [Research Report] RR-4848, INRIA. 2003. https://hal.inria.fr/inria-00071735/document

Page 1676, 1st paragraph

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    $\begingroup$ Typesetting note: in MathJAX (or LaTeX), the triangular brackets are created by the \langle and \rangle commands, not by the "less" and "greater" inequality signs. $\endgroup$ – zipirovich Jul 30 '18 at 22:12
  • $\begingroup$ I fixed that, thanks! $\endgroup$ – mueller_seb Jul 30 '18 at 22:33
  • $\begingroup$ You mention a source. What source would that be? Also, i don't know what an eigenvalue of a quadratic form would be. $\endgroup$ – Will Jagy Jul 31 '18 at 3:05
  • $\begingroup$ I added the source to the footer of my intro post now. I don't know any definition of these eigenvalues neither, only eigenvalues of the corresponding matrices. $\endgroup$ – mueller_seb Jul 31 '18 at 11:28
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I think that I solved the problem: The expression "eigenvalues of the quadratic form" mentioned in the source is simply not exact, but assuming these as eigenvalues of the matrix $B$ I defined above directly leads to the consequential inequality in the paper what is sufficient for my purposes. Due to the symmetry of $B$ it's diagonizable, so there exist an orthogonal matrix $S$ $(S^T S = I)$ such that $B = S^T D S$ with $D$ containing the eigenvalues $\lambda_i$. Then we get

$[x,Bx] = [x,S^T D S x] = [Sx, DSx] = \sum_i \lambda_i [Sx, Sx] \geq \min\{\lambda_i\} (\|x_1\|^2 + \|x_2\|^2)$

Thanks to all!

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