An extremal sparsity property of the Jordan canonical form (Q952031)

The paper deals with a particular property of the Jordan canonical form of a complex matrix. Given a complex matrix \(A\), of size \(n \times n\), \(J(A)\) denotes the Jordan canonical form of \(A\), \({\mathcal S}(A)\) the set of all complex matrices that are similar to \(A\) and \(\sigma(A)\) the number of off-diagonal nonzero entries of \(A\). The authors show that for any complex matrix \(A\), \(J(A)\) has the largest number of off-diagonal zero entries among all the matrices in \({\mathcal S}(A)\), and they characterize the matrices in \({\mathcal S}(A)\) that attain this largest number. Specifically, they prove that \[ \sigma(B) \geq \sigma(J(A)), \quad\text{for all } B \in {\mathcal S}(A), \] and the equality holds if and only if there exists a monomial matrix \(M\) (it has exactly one nonzero entry in each row and each column) such that \[ M^{-1}BM = J(A). \] Finally, the authors establish that a similar conclusion for the real Jordan canonical form does not hold. That is, if \(A\) is a real square matrix, the real Jordan canonical form of \(A\) has not the minimum number of off-diagonal nonzero entries among all real matrices in \({\mathcal S}(A)\).