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Sylvester's theorem or the Sylvester theorem may refer to any of several theorems named after James Joseph Sylvester:
- The Sylvester–Gallai theorem, on the existence of a line with only two of n given points.
- Sylvester's determinant identity, stating that det(I + AB) = det(I + BA), for matrices A, B.
- Sylvester's matrix theorem, also called Sylvester's formula, for a matrix function in terms of eigenvalues.
- Sylvester's law of inertia, also called 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.
- Sylvester's criterion, a characterization of positive-definite Hermitian matrices.
- Sylvester’s inequality about the rank (linear algebra) of the product of two matrices.
- Sylvester's closed solution for the Frobenius coin problem when there are only two coins.
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