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When dealing with quadratics, completing the square is ubiquitous, and I can summarise my interpretation of it as the formula:

$$x^2-2ax=(x-a)^2-a^2$$

Likewise, when working with circles (and, more generally, conic sections) in $\mathbb{R}^2$, it's customary to complete the square in a similar way. Sometimes this passes over to $\mathbb{C}$, for example, when computing the images of circlines under Mobius maps. In these circumstances, the equivalent formula typically manifests in the form:

$$\vert z \vert^2-a\bar z-\bar a z=\vert z-a \vert ^2 - \vert a \vert^2$$

Furthermore, when dealing with hyperbolas in 2D, as well as often in number theory, there's a similar type of rearrangement which arises and can be quite useful - I don't know a great way of describing it without an example, but consider:

$$xy+ax+by=(x+b)(y+a)-ba$$

Each of these effectively works on the basis of writing an algebraic expression as a canonical form minus a constant. I've found them collectively useful, and wanted to ask:

  • Would it be fair / understood to refer to the second of these techniques as completing the square? I'm sure that it does reflect the same principles as completing the square in 2D, but I'm not sure whether it's standard to refer to it in this way.

  • For the third of these techniques, is there an accepted name / standard way of referring to it? I do think of it as a spiritual successor to completing the square in some ways, but as there's no square involved, that seems like folly. Completing the rectangle perhaps?

  • Are there other common techniques that are comparable to the ones listed above, and is there a collective name for this type of approach? In extremely broad terms, I suppose it could be called putting expressions into a canonical form, but this doesn't really capture the essence of why these are special/distinguished.

πr8
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    One generic name is diagonalization. – André Nicolas Feb 08 '16 at 15:50
  • do you need a hilbert-spaces tag? – gt6989b Feb 08 '16 at 15:50
  • not sure, i would let it go, but not a strong opinion... let someone deeper comment on that and see where it goes... – gt6989b Feb 08 '16 at 15:57
  • @AndréNicolas I think that might be close? I think of diagonalisation as writing a matrix $M$ as $P^{-1}AP$, whereas from a more group-theoretic perspective, completing the square is writing a quadratic $q(x)$ as a translation of a squaring of a translation - I suppose $q=t_1 \circ s \circ t_2$, where $s$ is the squaring function and $t_1,t_2$ are translations. Perhaps a little reminiscent of singular value decompositions in that vein? – πr8 Feb 08 '16 at 16:05

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Your summary on completing square is excellent.

Each person may have different angle to look at the extension of completing square from their own knowledge. I would see the convex function is a further extension of completing square. The matrix (or operator) has extended it as the positive definite matrix (also convex function). Geometrically, A set can be separated by a line or plane from the rest of space. My view is limited by my own knowledge.

runaround
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