,j] the following: replace the powers ajk by ajk in the polynomiala1a2⋯an∏i<j(aj−ai),{\displaystyle a_{1}a_{2}\cdots a_{n}\prod _{i<j}(a_{j}-a_{i})\;,}where Π denotes the product of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real.[92] Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above; Kronecker's Vorlesungen über die Theorie der Determinanten[93] and Weierstrass' Zur Determinantentheorie,[94] both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.Many theorems were first established for small matrices only, for example the Cayley–Hamilton theorem was proved for 2×2 matrices by Cayley in the afor