In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin:

$$x^{T}x = ||{x}||^2$$ which represents the square of the Euclidean distance of the state from the equilibrium $x=0$.

So far, so good this is basic linear algebra. Then they go on to say:

In the following discussion, we will employ as a "generalized distance function" the quadratic form given by $${x}^TPx , P={P}^T $$ where $P$ is a real $n\times n$ matrix.

I am familiar with this definition of a quadratic form from linear algebra : we interpose a symmetric matrix to weight the variables in different ways.

But I am not familiar with this as a distance function. Is that mainly to say that it satisfies the requirements of a metric space? Is there a geometric (intuitive) discussion of the sense in which the quadratic form generalizes the more basic notion of a distance?

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    $\begingroup$ In general, it might not satisfy the requirements of a metric: unless $P$ is positive definite, distance between $x$ and $y \neq x$ may be zero or negative. $\endgroup$ – Budenn Jul 23 '15 at 20:03
  • $\begingroup$ @neuronet: Thanks for that post, it helped me with a Problem I have. Would you mind taking a look at my question regarding this Topic? I cant find a way to clearify this to me... math.stackexchange.com/questions/1634485/… $\endgroup$ – Benvorth Jan 31 '16 at 15:39
  • $\begingroup$ @Benvorth I still don't feel I grasped this problem yet, but see my comment to the answer below. The wikipedia page it links to I find largely impenetrable, unfortunately. I think a close study of mahalanobis distance is the way to go, as that has been addressed a lot more, at an intuitive level, and is pretty much a special case of this. I just haven't had time to undertake it yet. $\endgroup$ – neuronet Jan 31 '16 at 16:32

This determines a norm $\|\_\|_P$ iff $P$ is positive definite, then it naturally defines a metric by $d(x,y):=\|y-x\|_P$.

Else the above $d$ function would not be defined or would fail to be a metric, as e.g. there could be $x\ne 0$ with $x^TPx\le0$.

All in all, what it basically says is that the quadratic form $x\mapsto x^TPx$ can be viewed as a generalisation of $x\mapsto \|x\|^2$.

For more details and geometric insights, see for example Pseudo Euclidean spaces.

  • $\begingroup$ That is helpful, I wonder if there is a good discussion of this in a linear algebra context? That wikipedia page is tantalizing if a bit general, its connection to this specific measure gets lost to me fairly quickly. I am starting to think that the mahalanobis distance may be a good route to massage intuitions, as it seems to have been fairly well explored and is the same form as the quadratic form. $\endgroup$ – neuronet Jul 24 '15 at 15:08

I think some examples from physics might help provide the geometric (intuitive) sense you seek, in which quadratic forms generalize distance, though I doubt whether it’s useful to think of quadratic forms as providing a “more basic notion of distance” in quite the way that I think you're expecting. I see them as providing measures of distance like the usual (Pythagorean/Euclidean) notion of distance, but applied in slightly more general spaces.

A simple example of a generalisation of distance encoded by a quadratic form is given by the Minkowski metric for space-time; for the squared length of a vector from the origin:

$$d^2 = x^2 + y^2 + z^2 –(ct)^2$$

… which I’m sure you can see can be expressed as $X^TPX$ where $P$ is a $4\times 4$ matrix with all off-diagonal elements zero and the diagonal $\{1,1,1,-1\}$. See also http://mathworld.wolfram.com/MinkowskiMetric.html where the matrix is illustrated.

And so for the slightly less-clear case of a distance between the endpoints of vectors v1 and v2, I think you will now see this also as a quadratic form (using an overly-simplistic notation):

$$d^2(v1-v2) = (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 –(ct1-ct2)^2$$

In General Relativity, however, the matrix $P$ is called the Metric Tensor, and can have all non-zero components. As these components can change from point to point, this metric provides for a quadratic form that can encode distances in curved space-times, which is a generalisation of the "flat" Minkowski space-time.

So, as Berci pointed out via https://en.wikipedia.org/wiki/Pseudo-Euclidean_space , there’s a clear geometric interpretation, but it’s “simply” a generalisation of the familiar types of distances that apply in Pythagorean/Euclidean/pseudo-Euclidean spaces, rather than a generalisation of the more “disjointed” type that would be needed to include, say, Manhattan, Chessboard or Mahalanobis distances.

(Clearly, I’m not a mathematician, but I think the physics examples help provide the geometric insight you seek, and if I’m wrong in this, would appreciate correction from those more expert. Put into MathJax with assistance acknowledged below.)

  • $\begingroup$ Appreciate the formatting edit, Siong Thye Goh. . $\endgroup$ – iSeeker Oct 1 '17 at 23:22

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