Isospectral
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In mathematics, two linear operators are called isospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. In the case of operators on infinite-dimensional spaces, the spectrum need not consist solely of isolated eigenvalues; the rest of this article will assume for clarity that we are talking about operators on finite-dimensional vector spaces.
In that case, for complex square matrices, the relation of being isospectral for two diagonalizable matrices is just 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)−1AM(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 ′ = [A, M] = AM − 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 pairs (P,L) giving rise to analogous equations, by Peter Lax, showed how linear machinery could explain the non-linear behaviour.
