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Expected Pseudospectrum Area of Random Haar Compressions of Matrices

Non-asymptotic upper bounds on the expected area of the epsilon-pseudospectrum of a compression Q*AQ of a fixed matrix A onto a random subspace drawn from the Haar measure on the complex Grassmannian. The bounds scale like poly(n) log^2(1/epsilon) times epsilon^beta with beta in {6/5, 4/3, 2} depending on the numerical range and singular-value gaps of A, showing random compressions of highly non-normal matrices keep their eigenvalues far more stable than the worst case.

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The Pseudospectrum of Random Compressions of Matrices Rikhav Shah∗ UC Berkeley January 3, 2025

Studies the pseudospectrum -- the set of eigenvalues of all matrices lying within operator-norm distance epsilon of a given matrix, whose area measures how unstable the eigenvalues are -- of a random…