I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution to the topic? What is the origin and the history of methods of the investigation of spectral properties of Jacobi matrices?
Any suitable reference concerning the above questions would be helpful. Thanks.
In 1845 paper http://www.ima.umn.edu/preprints/April92/951.pdf (On a New Way of Solving the Linear Equations that Arise in the Method of Least Squares) Jacobi introduced a new iterative method to solve the matrix equation $Ax=b$, where the matrix $A$ is diagonally dominant. As an example, he considers the case $$A=\left(\begin{array}{ccc} 27 & 6 & 0 \\ 6 & 15 & 1 \\ 0 & 1 & 54\end{array} \right),\;\;\;b=\left(\begin{array}{c} 88 \\ 70 \\ 107\end{array}\right).$$ Jacobi uses a rotation to eliminate the biggest off-diagonal element $A_{12}= A_{21}=6$ and then solves the transformed system in three iterations each adding about one digit of accuracy. Perhaps this was a starting point of the theory of tridiagonal (Jacobi) matrices.
In 1950, Lanczos introduced a method for the successive transformation of a given matrix to a tridiagonal matrix which turned to be very important for solving linear systems of equations and eigenvalue problems. For historical overview of these developments see http://www.cs.umd.edu/~oleary/reprints/j28.pdf (Some history of the conjugate gradient and Lanczos algorithms : 1948-1976, by Gene H Golub and Dianne P O'Leary).
Some historically important references about the spectral theory of Jacobi operators can be found in http://annals.math.princeton.edu/wp-content/uploads/annals-v158-n1-p05.pdf (Sum rules for Jacobi matrices and their applications to spectral theory, by Rowan Killip and Barry Simon).
