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# Isospectral

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.

The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices. Two diagonalizable matrices are isospectral just when they are similar.

In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the shape of a drum.

## Finite dimensional spaces

In the case of operators on finite-dimensional vector spaces, 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] = AMMA

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.

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