Almost Commuting Matrices Need not be Nearly Commuting

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.

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...

Commuting triples of matrices

The variety C(3, n) of commuting triples of n × n matrices over C is shown to be irreducible for n = 7. It had been proved that C(3, n) is reducible for n ≥ 30, but irreducible for n ≤ 6. Guralnick and Omladič have conjectured that it is reducible for n > 7.

Almost Commuting Unitaries with Spectral Gap Are near Commuting Unitaries

Let Mn be the collection of n×n complex matrices equipped with operator norm. Suppose U, V ∈ Mn are two unitary matrices, each possessing a gap larger than ∆ in their spectrum, which satisfy ‖UV −V U‖ ≤ ǫ. Then it is shown that there are two unitary operators X and Y satisfying XY −Y X = 0 and ‖U−X‖+‖V −Y ‖ ≤ E(∆/ǫ) “ ǫ ∆2 ” 1 6 , where E(x) is a function growing slower than x 1 k for any posit...