Making Almost Commuting Matrices Commute

Almost-commuting matrices are almost jointly diagonalizable

We study the relation between approximate joint diagonalization of self-adjoint matrices and the norm of their commutator, and show that almost commuting self-adjoint matrices are almost jointly diagonalizable by a unitary matrix.

Making Laplacians Commute: Multimodal Spectral Geometry Using Closest Commuting Operators∗

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. We show that our problem is equivalent to joint diagonalization, however, unlike joint diagonalization, allows to explicitly preserve the properties of Laplacians. Our construction n...

Representation spaces for central extensions and almost commuting unitary matrices

On the Variety of Almost Commuting Nilpotent Matrices

Let V be a vector space of dimension n over a fieldK of characteristic equal to 0 or ≥ n/2. Let g = gln(V ) and n be the nilcone of g, i.e., the cone of nilpotent matrices of g. We write elements of V and V ∗ as column and row vectors, respectively. In this paper we study the variety N := {(X,Y, i, j) ∈ n× n × V × V ∗ | [X,Y ] + ij = 0} and prove that it has n irreducible components: 2 of dimen...

Making Laplacians commute

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real exa...