A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi.

The name derives from a theorem from Jacobi, dating to 1848, stating that every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix.

The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers ${\displaystyle \ell ^{2}(\mathbb {N} )}$. In this case it is given by

where the coefficients are assumed to satisfy

The operator will be bounded if and only if the coefficients are bounded.

There are close connections with the theory of orthogonal polynomials. In fact, the solution ${\displaystyle p_{n}(x)}$ of the recurrence relation

is a polynomial of degree n and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector ${\displaystyle \delta _{1,n}}$.

This recurrence relation is also commonly written as