Talk:Orthogonality

From Academic Kids

Those trained in computer science think they invented everything known before computers existed: integrals, mathematical induction, orthogonality, etc. I've left the page a bit of a messy hodge-podge, but far better than what was here. Michael Hardy 02:27, 13 Jan 2004 (UTC)

If non-orthodox is "heterodox", is "heterogonal" non-orthogonal? (Google has one hit for that word (http://www.google.com/search?q=heterogonal), in an unmaths context.) 142.177.126.230 21:05, 5 Aug 2004 (UTC)

It is a needless complication of the definition of orthogonality to bring in the subscripts i and j when one is only trying to define what it means to say two functions are orthogonal. And it is incorrect unless one has first given them some meaning. Michael Hardy 01:45, 6 Sep 2004 (UTC)

Contents

1 examples 2 Missing bracket 3 vectors 4 Orthogonal curves

examples

i'd like an example of two simple functions that are orthogonal. - Omegatron 16:22, Sep 29, 2004 (UTC)

Take two orthogonal vectors and then change basis to {1, t, t^2, ..., t^n}?

Dysprosia 22:32, 29 Sep 2004 (UTC)

No, that won't work until you specify a measure (or "weight function") with respect to which those are orthogonal. See for example Chebyshev polynomials and Legendre polynomials and Hermite polynomials (all exceptions to the rule that it is better use singulars as Wikipedia article titles). Those are examples. Also, see Bessel function. Michael Hardy 00:32, 30 Sep 2004 (UTC)

Well, it does depend on the inner product you use to determine orthogonality, though. But yes, if you use the inner product defined in the article, it won't work. Dysprosia 01:48, 30 Sep 2004 (UTC)

some of us don't know what that means... aren't sin and cosine orthogonal? and certain pulse trains? - Omegatron 22:53, Sep 29, 2004 (UTC)

If you use the inner product from the article, and take the integral from -a to a with weight 1, sin x and cos x are indeed orthogonal (calculate it for yourself). Dysprosia 01:48, 30 Sep 2004 (UTC)

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And explain why the integral is a to b instead of -∞ to +∞? - Omegatron 16:24, Sep 29, 2004 (UTC)

No reason, though you can define another inner product with those bounds and then consider orthogonality with respect to that inner product.Dysprosia 22:32, 29 Sep 2004 (UTC)

i see that the a and b are used in the inner product article, too. - Omegatron 22:53, Sep 29, 2004 (UTC)

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Missing bracket

There is a missing opening square bracket on the integration example image, I believe. --anon

Fixed now. I think that bracket was left out on purpose. But I agree with you that things look better with the bracket in. Oleg Alexandrov 18:25, 15 May 2005 (UTC)

vectors

for some positive integer a, and for 1 ≤ k ≤ a-1, these vectors are orthogonal, for example (1,0,0,1,0,0,1,0)^T,(0,1,0,0,1,0,0,1)^T ,(0,0,1,0,0,1,0,0)^T are orthogonal.

interesting. so this is where discretely sampled signals like

...0,0,1,0,0,1,1...
...1,0,0,0,1,0,0...
...0,1,0,1,0,0,0...

come from? and these signals are orthogonal too, according to another site I saw. can we extrapolate the signal processing version from the many dimensional vector version? maybe graphs? - Omegatron 13:41, Sep 30, 2004 (UTC)

They appear to be. Calculate the dot product of these "signals", so to speak, across each triplet. If they sum to 0 for all the bit triplets over your time period they are orthogonal. I don't understand what you mean about "extrapolate the signal processing version from the many dimensional vector version". Dysprosia 14:04, 30 Sep 2004 (UTC)

the difference being that this is a discrete function instead of a vector,

function <math>f[n] = ...,0,1,0,0,4,0,0,-1,0,2,...<math>

vector <math>\mathbf{a} = (...,0,1,0,0,4,0,0,-1,0,2,...)<math>

but i guess they can be seen as the same thing from different perspectives? can you have infinite-dimensional vectors? the discrete-"time" function can be "converted" to a continuous-time function (think sampling), though, which can also be orthogonal to another similar function if they have the same "shape" relationship... - Omegatron 14:40, Sep 30, 2004 (UTC)

heh. lots of "quotes". i can explain better later. i will draw some pictures... - Omegatron 14:41, Sep 30, 2004 (UTC)

Yes, you can have vectors of infinite dimension. You know there is in fact nothing really special about any of these definitions of orthogonality - what is the important property is the inner product, which determines whether two vectors in a vector space are orthogonal or not, or determines a "length" or not. Change the inner product, and these definitions change also. Dysprosia 14:49, 30 Sep 2004 (UTC)

Not sure I understand what you're trying to say. So you could define your own "inner product" for which a cat is orthogonal to a dog? - Omegatron 19:55, Sep 30, 2004 (UTC)

Metaphorically, yes, as long as the inner product you define is in fact an inner product. There are some requirements on this, see inner product. Literally, you have to define what you mean by a cat and dog first before you can say they are orthogonal to each other... ;) Dysprosia 01:07, 1 Oct 2004 (UTC)

can you have infinite-dimensional vectors?

Except that it's the space that is infinite-dimensional, rather than the vectors themselves. The two most well-known infinite-dimensional vector spaces are <math>\ell^2<math>, which is the set of all sequences of scalars such that the sum of the squares of their norms is finite (for example (1, 1/2, 1/3, ...) is such a vector because 1^{2</sub> + (1/2)2 + (1/3)2 + ... is finite) and L₂, the set of all functions f such that}

<math>\int_\mathrm{whatever\ space}\left|f\right|^2 < \infty.<math>

("Whatever space" could be for example the interval from 0 to 2π, or could be the whole real line, or could be something else.) Michael Hardy 19:30, 30 Sep 2004 (UTC)

Yes. So what is the connection between the discrete function with an infinite number of points ...,f[-1],f[0],f[1],... and a vector with an infinite number of dimensions (...,x_-1,x₀,x₁,...)? Are these the same concept said in two different ways or are there subtle differences? For instance, in MATLAB or GNU Octave you use vectors or matrices for everything, and use them to represent strings of sampled data or two dimensional arrays of data, both of which could also be thought of as functions of the vector or matrix coordinates.

Not that this is a site for teaching people math, but it could point out things that need to be included in various articles.  :-) - Omegatron 19:55, Sep 30, 2004 (UTC)

Let x_i = f(i)? Dysprosia 01:07, 1 Oct 2004 (UTC)

Orthogonal curves

This article does not mention orthogonal curves or explain what it means that two circles are orthogonal to each other. Hyperbolic geometry mentions orthogonal circles, but I had to look up the exact meaning elsewhere (more precisely, on MathWorld).

My question is, should orthogonal curves and circles be covered in this article, or do they qualify as a "related topic"? Fredrik | talk 03:16, 21 Oct 2004 (UTC)

The concept's not really that different, though Mathworld's geometric treatment may merit a seperate page. One could perhaps say generally that two curves parametrized by functions f and g are orthogonal, if where they interesect ∇f.∇g = 0, though I'm not sure that's a decent, established, or useful definition... Dysprosia 08:14, 21 Oct 2004 (UTC)