Sylvester's theorem
Encyclopedia
Sylvester's theorem or Sylvester theorem may refer to any of several theorems named after James Joseph Sylvester
:
It may also refer to the following theorems proved by J. J. Sylvester:
James Joseph Sylvester
James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
:
- The Sylvester–Gallai theoremSylvester–Gallai theoremThe Sylvester–Gallai theorem asserts that given a finite number of points in the Euclidean plane, either# all the points are collinear; or# there is a line which contains exactly two of the points.This claim was posed as a problem by...
, on the existence of a line with only two of n given points. - Sylvester's determinant theoremSylvester's determinant theoremIn matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants. It is named after James Joseph Sylvester....
, stating that det(I + AB) = det(I + BA), for matrices A, B. - Sylvester's matrix theorem, a.k.a. Sylvester's formulaSylvester's formulaIn matrix theory, Sylvester's formula or Sylvester's matrix theorem expresses an analytic function f of a matrix A in terms of the eigenvalues and eigenvectors of A...
, for a matrix function in terms of eigenvalues. - Sylvester's law of inertiaSylvester's law of inertiaSylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...
a.k.a. Sylvester's rigidity theorem, about the signature of a quadratic form. - Sylvester's theorem on the product of k consecutive integers > k, that generalizes Bertrand's postulate.
- Sylvester's theorem on partitions.
- Sylvester theorem on spherical harmonics.
It may also refer to the following theorems proved by J. J. Sylvester:
- Sylvester's criterionSylvester's criterionIn mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester....
, a characterization of positive-definite Hermitian matrices. - Sylvester’s inequality about the rank (linear algebra)Rank (linear algebra)The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
of the product of two matrices. - Sylvester's closed solution for the Frobenius coin problemCoin problemThe coin problem is a mathematical problem that asks what is the largest monetary amount that cannot be obtained using only coins of specified denominations. For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units...
when there are only two coins.