Narrow Results

Given a rational matrix A, and a set of rational matrices B, C,… which commute with A, we give polynomial time algorithms to compute exactly the Jordan Normal Form of A, as well as the transformed matrices of B, C,…. We also obtain the transformation matrix and its inverse exactly in polynomial time.

We establish that the problem of computing the Jordan normal form of a matrix over a field F is in for F being a field of characteristic zero or a finite field.

This paper studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of ${\mathrm{\Psi }}_{\epsilon }$-associated consistency, which require that a value is invariant under some parametrized linear transformation ${\mathrm{\Psi }}_{\epsilon }$ on the vector space of cooperative games with transferable utility. Xu et al. [(2008) Linear Algebr. Appl.428, 1571–1586; (2009) Linear Algebr. Appl.430, 2896–2897] Xu et al. [(2013) Linear Algebr. Appl.439, 2205–2215], Hamiache [(2010) Int. Game Theor. Rev.12, 175–187] and more recently Xu et al. [(2015) Linear Algebr. Appl.471, 224–240] follow this approach by using a matrix analysis. The main drawback of these articles is the heaviness of the proofs to show that the matrix expression of the linear transformations is diagonalizable. By contrast, we provide quick proofs by relying on the Jordan normal form of the previous matrix.