Majorization for Changes in Angles Between Subspaces, Ritz Values, and Graph Laplacian Spectra

Abstract

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ${\prec}_{w}$. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let $\Theta({\mathcal X},{\mathcal Y})$ be the vector of principal angles in nondecreasing order between subspaces ${\mathcal X}$ and ${\mathcal Y}$ of a finite dimensional space ${\mathcal H}$ with a scalar product. We consider the change in principal angles between subspaces ${\mathcal X}$ and ${\mathcal Z}$, where we let ${\mathcal X}$ be perturbed to give ${\mathcal Y}$. We measure the change using weak majorization. We prove that $|\cos^2\Theta({\mathcal X},{\mathcal Z})-\cos^2\Theta({\mathcal Y},{\mathcal Z})| {\prec}_{w} \sin\Theta({\mathcal X},{\mathcal Y})$, and give similar results for differences of cosines, i.e., $|\cos\Theta({\mathcal X},{\mathcal Z})-\cos\Theta({\mathcal Y},{\mathcal Z})| {\prec}_{w} \sin\Theta({\mathcal X},{\mathcal Y})$, and of sines and sines squared, assuming $\dim {\mathcal X} = \dim {\mathcal Y}$. We observe that $\cos^2\Theta({\mathcal X},{\mathcal Z})$ can be interpreted as a vector of Ritz values, where the Rayleigh–Ritz method is applied to the orthogonal projector on ${\mathcal Z}$ using ${\mathcal X}$ as a trial subspace. Thus, our result for the squares of cosines can be viewed as a bound on the change in the Ritz values of an orthogonal projector. We then extend it to prove a general result for Ritz values for an arbitrary Hermitian operator A, not necessarily a projector: let $\Lambda(P_{{\mathcal X}}A|_{{\mathcal X}})$ be the vector of Ritz values in nonincreasing order for A on a trial subspace ${\mathcal X}$, which is perturbed to give another trial subspace ${\mathcal Y}$; then $| \Lambda(P_{{\mathcal X}}A|_{{\mathcal X}})- \Lambda(P_{{\mathcal Y}}A|_{{\mathcal Y}})|\prec_w (\lmax-\lmin)~\sin\Theta({\mathcal X},{\mathcal Y})$, where the constant is the difference between the largest and the smallest eigenvalues of A. This establishes our conjecture that the root two factor in our earlier estimate may be eliminated. Our present proof is based on a classical but rarely used technique of extending a Hermitian operator in ${\mathcal H}$ to an orthogonal projector in the “double” space ${\mathcal H}^2$. An application of our Ritz values weak majorization result for Laplacian graph spectra comparison is suggested, based on the possibility of interpreting eigenvalues of the edge Laplacian of a given graph as Ritz values of the edge Laplacian of the complete graph. We prove that $|\cos^2\Theta({\mathcal X},{\mathcal Z})-\cos^2\Theta({\mathcal Y},{\mathcal Z})| {\prec}_{w} \sin\Theta({\mathcal X},{\mathcal Y})$ where $\lambda^1_k$ and $\lambda^2_k$ are all ordered elements of the Laplacian spectra of two graphs with the same n vertices and with l equal to the number of differing edges.