Theorem 1 (Eigenvector-eigenvalue identity) Let be an Hermitian matrix, with eigenvalues . Let be a unit eigenvector corresponding to the eigenvalue , and let be the component of . Then where is the Hermitian matrix formed by deleting the row and column from . When we posted the first ve...
is an affine rescaling to the scale of the eigenvalue gap. So matters soon reduce to controlling the probability of the event where is the number of eigenvalues to the right of , and is the number of eigenvalues in the interval . These are fixed energy events, and one can use the the...
These operators have a joint spectrum\(\sigma (\underline{a})\), whose elements are the joint eigenvalues\(\underline{\lambda }=(\lambda _1, \ldots , \lambda _N)\), defined by the property that there exists a nonzero joint eigenvector \(\psi \in K\) such that \(a_i\psi =\...
Our focus is on vector-valued data sources, each consisting of measurements of the same entity but on different variables, and on tasks where source-specific variation is considered noisy or not interesting. Principal components analysis of all sources combined together is an obvious choice if it ...
Fundamentally a FRF is a mathematical representation of the relationship between the input and the output of a system. So for example the FRF between two points on a structure. It would be possible to attach an accelerometer at a particular point and excite the structure at another point with...
The Smith normal form takes an arbitrary matrix and factorises it as , where , , and is a rectangular diagonal matrix, by which we mean that the principal minor is diagonal, with all other entries zero. Furthermore the diagonal entries of are for some (which is also the rank of ) ...
For GUE, which is a continuous matrix ensemble, one can also define for distinct as the unique quantity such that the probability that there is an eigenvalue in each of the intervals is in the limit . As is well known, the GUE process is a determinantal point process, which means that ...
So it is important to gather “enough” inequalities so that the relevant matrix has a Perron-Frobenius eigenvalue greater than or equal to (and in the latter case one needs non-trivial injection of an induction hypothesis into an eigenspace corresponding to an eigenvalue ). More specifically,...
As explained in Section 1.2, this plane does not need to be necessarily fully invariant, but it should be at least unrotated by the distortion [48]—i.e., it should be an eigenvector of the distortion matrix expressed in the reciprocal space 𝐅∗=𝐅−tF*=F−t. Combining two ...
As explained in Section 1.2, this plane does not need to be necessarily fully invariant, but it should be at least unrotated by the distortion [48]—i.e., it should be an eigenvector of the distortion matrix expressed in the reciprocal space 𝐅∗=𝐅−tF*=F−t. Combining two ...