Enhancing the Jordan canonical form

The Jordan canonical form parametrises similarity classes in the nilpotent cone Nn, consisting of n× n nilpotent complex matrices, by partitions of n. Achar and Henderson (2008) extended this and other well-known results about Nn to the case of the enhanced nilpotent cone C ×Nn. 1. Jordan canonical form The Jordan canonical form (JCF), introduced in 1870 [10], is one of the most useful tools in linear algebra. As an illustration, consider the following result. (Here and throughout, all matrices have entries in C.) Proposition 1.1. Let A be an invertible n×n matrix. Then A has a square root: that is, there exists an n× n matrix B such that B = A. The n = 1 case of Proposition 1.1 just says that any nonzero complex number a has a square root √ a, which is one of the well-known advantages of working in C. The n = 2 case is already a bit tricky, if we use only the definition of matrix multiplication: it amounts to showing that there is a solution to a system of four degree-2 equations in the four unknown entries of B. (This is not automatic, as shown by the existence of noninvertible 2× 2 matrices with no square root.) Attempting to prove the n = 3 case this way would be foolish. A better approach to Proposition 1.1 is the maxim ‘use the symmetry of the problem to reduce to a special case’. Remember that n× n matrices A and A are said to be similar if A = XAX for some invertible matrix X. If this is the case, then A has a square root if and only if A has a square root, because of the easy observation that (XBX) = XBX . This means that we only need to consider one representative A from each similarity class of n× n invertible matrices. The JCF theorem gives us such a representative. Theorem 1.1 (Jordan canonical form). Every similarity class of n× n matrices contains a matrix A that is block-diagonal with diagonal blocks J`1(a1), J`2(a2), . . . , Invited technical paper, communicated by Jon Borwein. School of Mathematics and Statistics, University of Sydney, NSW 2006. Email: [email protected] Enhancing the Jordan canonical form 207 J`k(ak) for some `i ∈ Z and ai ∈ C, where