The term "Jacobi matrix" has several meanings in mathematics.
One meaning is the matrix of first partial derivatives of a vector function, also called a Jacobian. The Jacobian determinant is its determinant when the matrix is square.
The term is also used for a Jacobi rotation matrix.
Another meaning is a tridiagonal matrix that encodes the three-term recurrence relation for a family of orthogonal polynomials. For orthogonal polynomials normalized as orthonormal functions with positive leading coefficients, the Jacobi matrix is the real symmetric matrix
![J=[b_0 a_1 0 0 ...; a_1 b_1 a_2 0 ...; 0 a_2 b_2 a_3 ...; | | | ... ...],](/images/equations/JacobiMatrix/NumberedEquation1.svg) |
(1)
|
where 
is real and 
for 
. The corresponding orthonormal polynomials satisfy
 |
(2)
|
where 
and 
. With a different normalization, the two off-diagonal
sequences need not be equal. Favard's theorem
gives the positive measure associated with these
coefficients.
Let 
be the leading 
block of 
. If 
is the monic polynomial
proportional to 
, then
 |
(3)
|
Thus the eigenvalues of 
are the zeros of 
. They are real and
distinct, and the zeros of consecutive orthogonal
polynomials interlace. For unit eigenvectors 
corresponding to eigenvalues 
, the weights of Gaussian
quadrature are
 |
(4)
|
where 
is the total mass of the measure. This is the matrix
method of Golub and Welsch (1969, pp. 221-223).
