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The problem of computing a p-dimensional invariant subspace of a symmetric positive-definite matrix pencil of dimension n is interpreted as computing a zero of a tangent vector field on the Grassmann manifold of p-planes in Rn. The theory of Newton’s method on manifolds is applied to this problem, and the resulting Newton equations are interpreted as block versions of the Jacobi–Davidson correction equation for the generalized eigenvalue problem.

The problem of computing a p-dimensional invariant subspace of a symmetric positive-definite matrix pencil of dimension n is interpreted as computing a zero of a tangent vector field on the Grassmann manifold of p-planes in R^n. The theory of Newton's method on manifolds is applied to this problem, and the resulting Newton equations are interpreted as block versions of the Jacobi-Davidson correction equation for the generalized eigenvalue problem.

Newton's method for solving the matrix equation $F(X)\equiv AX-XX^TAX=0$ runs up against the fact that its zeros are not isolated. This is due to a symmetry of $F$ by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a ``geometric'' Newton algorithm that finds the zeros of $F$. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method.

Higher-order tensors have applications in many areas such as biomedical engineering, image processing, and signal processing. For example, dimensionality reduction of a multi-way problem can be achieved by the best rank-(R 1 ,R 2 ,. . . ,R N) approximation of tensors. Contrary to the matrix case, the tensor best rank-(R 1 , R 2 , . . . , R N) approximation cannot be computed in a straightforward way. In this paper, we present the higher-order orthogonal iter-ations and outline two new algorithms, based on the trust-region and conjugate gradient methods on manifolds. We touch on some of the applications.