Editing Isospectral (section)

==Finite dimensional spaces==

In the case of operators on finite-dimensional vector spaces, for [[complex number|complex]] square matrices, the relation of being isospectral for two [[diagonalizable matrix|diagonalizable matrices]] is just [[similar (linear algebra)|similarity]]. This doesn't however reduce completely the interest of the concept, since we can have an '''isospectral family''' of matrices of shape ''A''(''t'') = ''M''(''t'')<sup>&minus;1</sup>''AM''(''t'') depending on a [[parameter]] ''t'' in a complicated way. This is an evolution of a matrix that happens inside one similarity class.

A fundamental insight in [[soliton]] theory was that the [[infinitesimal]] analogue of that equation, namely

:''A''{{prime}} = [''A'', ''M''] = ''AM'' &minus; ''MA''

was behind the conservation laws that were responsible for keeping solitons from dissipating. That is, the preservation of spectrum was an interpretation of the conservation mechanism. The identification of so-called [[Lax pair]]s (P,L) giving rise to analogous equations, by [[Peter Lax]], showed how linear machinery could explain the non-linear behaviour.