Introducing \( p\)-eigenvectors, exact solutions for
some simple matrices

Abstract. A common way to define a norm of a matrix is to take the supremum of the fraction of the vector norms of the matrix-vector product and the nonzero vector, with respect to a given vector norm, i.e. the least upper bound for the norm of the vectors of the transformed unit sphere. In this paper we examine the above mentioned fraction, defining induction curves and surfaces, we show that there exist some vectors, such that this fraction is independent of the applied \(p\)-norm (and are not eigenvectors). These are to be called \(p\)-eigenvectors. Exact solutions are constructed for some simple matrices. No previous work was found in this topic so far.