Finding Eigenvalues for Matrices Acting on Subspaces
Jakeniah Christiansen · SIAM Undergraduate Research Online · 2012
Consider the eigenvalue problem QAQ x = λ x, where A is an n×n matrix and Q is projection matrix onto a subspace S ⊥ of dimension n -k.In this paper we construct a meromorphic function whose zeros correspond to the (generically) n-k nonzero eigenvalues of QAQ.The construction of this function requires only that we know A and a basis for S, the orthogonal complement of S ⊥ .The formulation of the function is assisted through the use of the Frobenius inner product; furthermore, this inner product allows us to directly compute the eigenvalue when k = 1 and n = 2.When n = 3 and k = 1 we carefully study four canonical cases, as well as one more general case.