The rotation of eigenvectors by a perturbation

Accuracy of singular vectors obtained by projection-based SVD methods

“…As a historical sidenote, Theorem 2 (or more specifically the setup in (5)) is related to Question 10.2 in the classical paper by Davis and Kahan [4], where they ask for subspace angle bounds between three subspaces. Theorem 2 is related but not a direct answer to their question, since it provides bounds not on the direct subspace angles (as in Davis-Kahan's sin θ, tan θ theorems) but the relative quality of the extracted subspace sin (X 1 , X 1 ) / sin (X 1 , X ) , where dim X > dim X 1 .…”

Jordan's principal angles in complex vector spaces

“…The available definitions give only the principal angles and use CS decomposition [3,10] or eigenvalues (see, e.g. References [4,5,8,[10][11][12][13]). Hence there is some loss of the geometric character.…”

“…For example, if one defines cos = | x, y |/( x y ), then = /2 if and only if x, y = 0, but the law of cosines does not hold. If one uses cos = Re x, y /( x y ), then the law of cosines holds, but = /2 may hold for x, y = 0 (see also References [4,15]). The variety of angle definitions and properties clearly requires a thorough extension of Jordan's original concept [1,2] to the complex case.…”

“…Although Jordan [1] introduced the concept of principal angles (and vectors) in 1875 (see also References [2,3]), the principal angles were rediscovered several times. Davis and Kahan [4] include an interesting account of these works to which we can add Reference [5] as the most recent contribution.…”

A pointwise criterion for controller robustness

“…The study of angles between subspaces of a Euclidean space goes hack to Camille Jordan [20], who showed that the mutual position of two complementary subspaces in Euclidean space is characterized completely by a finite number of invariants which he called the angles between the subspaces. The subject has drawn interest not only from geometers [11,34] but also from operator theorists [16], numerical analysts (both as an analysis tool [6,7] and as a subject of computation [4,18] …”