Sum of squares

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In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:


Number theory[edit]

Algebra, algebraic geometry, and optimization[edit]

  • Polynomial SOS, polynomials that are sums of squares of other polynomials
  • The Brahmagupta–Fibonacci identity, representing the product of sums of two squares of polynomials as another sum of squares
  • Hilbert's seventeenth problem on characterizing the polynomials with non-negative values as sums of squares
  • Sum-of-squares optimization, nonlinear programming with polynomial SOS constraints
  • The sum of squared dimensions of a finite group's pairwise nonequivalent complex representations is equal to cardinality of that group.

Euclidean geometry and other inner-product spaces[edit]

  • The Pythagorean theorem says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs. The sum of squares is not factorable.
  • The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates
  • Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares)
  • The British flag theorem for rectangles equates two sums of two squares
  • The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals
  • Descartes' theorem for four kissing circles involves sums of squares
  • The sum of the squares of the edges of a rectangular cuboid equals the square of any space diagonal

See also[edit]