To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A, E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the solution can exhibit transient growth before its inevitable decay. When the equation results from the linearization of a nonlinear system, this transient growth gives a mechanism that can promote nonlinear instability. One can enrich the conventional large-scale eigenvalue calculation used for linear stability analysis to identify the potential for such transient growth. Toward this end, we introduce a new definition of the pseudospectrum of a matrix pencil, use it to bound transient growth, explain how to incorporate a physically-relevant norm, and derive approximate pseudospectra using the invariant subspace computed in conventional linear stability analysis. We apply these tools to several canonical test problems in fluid mechanics, an important source of differential-algebraic equations.

In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold M with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, E λ px, yq, of the projection operator from L 2 pM q onto the direct sum of eigenspaces with eigenvalue smaller than λ 2 as λ Ñ 8. We obtain a uniform logarithmic improvement in the remainder of this asymptotic expansion when the points x, y are close together. This result is a generalization of a work by Bérard, which treated the on-diagonal case, E λ px, xq. The results in this paper allow us to conclude that the rescaled covariance kernel of a monochromatic random wave on a manifold without conjugate points locally converges to a universal scaling limit at an inverse logarithmic rate.

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