Now, the statement (i) follows from (S.1.3) by setting y = x. (ii) Since the vectors ci1 , . . . , ciK−1 are linearly independent in R , it follows that there exists a vector y such that det [ ci1 . . . ciK−1 y ] 6= 0. Hence, by (S.1.3), the (i1, . . . , iK−1)-th column of B(C) is nonzero. (iii) follows from (S.1.3) and the fact that det [ ci1 . . . ciK−1 y ] = 0 if and only if y ∈ span{ci1 , ....

This paper provides algorithms for adding and subtracting eigenspaces, thus allowing for incremental updating and downdating of data models. Importantly, and unlike previous work, we keep an accurate track of the mean of the data, which allows our methods to be used in classification applications. The result of adding eigenspaces, each made from a set of data, is an approximation to that which ...

A method for simultaneous modelling of the Cholesky decomposition of several covariance matrices is presented. We highlight the conceptual and computational advantages of the unconstrained parameterization of the Cholesky decomposition and compare the results with those obtained using the classical spectral (eigenvalue) and variance-correlation decompositions. All these methods amount to decomp...

A tutorial is presented on the Proper Orthogonal Decomposition (POD), which finds applications in computationally processing large amounts of high-dimensional data with the aim of obtaining low-dimensional descriptions that capture much of the phenomena of interest. The discrete version of the POD, which is the singular value decomposition (SVD) of matrices, is described in some detail. The con...

More robust developments of schemes for hyperbolic systems, that avoid dependence upon a characteristic decomposition have been achieved by employing some form of a Lax-Friedrichs (LF)based flux. Such schemes permit the construction of higher order approximations without recourse to characteristic decomposition. This is achieved by using the maximum eigenvalue of the hyperbolic system within th...

Out-of-time-ordered correlators (OTOC) are a quantifier of quantum information scrambling and chaos. We propose an efficient algorithm to measure OTOCs that provides exponential speed-up over the best known classical provided OTOC operator be estimated admits gate decomposition. also discuss scheme obtain about eigenvalue spectrum spectral density OTOCs.