Matrix pencils and a generalized Clifford algebra

Abstract

Let F be an algebraically closed field, and x 1,…,x m be commuting indeterminates over F. For a given monic polynomial φ(z)∈F[x 1,…,x m][z] which is homogeneous (viewed as an element of F[x 1,…,x m,z] ), we construct an associative algebra C φ which we call the generalized Clifford algebra of the polynomial φ. This construction generalizes that of Roby's Clifford algebra, and is a universal algebra for the problem of finding matrices A 1,…,A m∈M n(F) for some n such that φ(z) is the minimal polynomial of the matrix pencil x 1A 1+⋯+x mA m . Our main result is that, if φ(z) is quadratic, then C φ is either a matrix algebra of dimension a power of 2 or a direct sum of two such matrix algebras, and we conclude that the problem of finding m matrices in M n(F) whose pencil has a prescribed quadratic minimal polynomial can be solved, if and only if n is an appropriate power of 2. We apply this result to the problem of bounding the lengths of generating sets for matrix algebras and discuss some of the difficulties encountered when the degree of φ(z) is ⩾3.