Matrix pencils and a generalized Clifford algebra Original Research Article

Let F be an algebraically closed field, and x1,…,xm be commuting indeterminates over F. For a given monic polynomial φ(z)set membership, variantF[x1,…,xm][z] which is homogeneous (viewed as an element of F[x1,…,xm,z]), we construct an associative algebra Cφ which we call the generalized Clifford algebra of the polynomial φ. This construction generalizes that of Robyʹs Clifford algebra, and is a universal algebra for the problem of finding matrices A1,…,Amset membership, variantMn(F) for some n such that φ(z) is the minimal polynomial of the matrix pencil x1A1+cdots, three dots, centered+xmAm. Our main result is that, if φ(z) is quadratic, then Cφ 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 Mn(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 greater-or-equal, slanted3.