Rajaeadot

Rajaeadot is a hypothetical mathematical construct used to illustrate a generalized bilinear form on a finite-dimensional real vector space. It is defined for vectors x and y in R^n by Rajaeadot(x, y) = x^T A y, where A is a fixed real n-by-n matrix, sometimes called the Rajaeadot kernel. The term blends the surname Rajaea with the mathematical dot product.

Properties: Rajaeadot is bilinear in its arguments. It is symmetric if A is symmetric, and positive definiteness occurs when A is symmetric positive definite, in which case Rajaeadot defines an inner product equivalent to the standard dot product up to a linear transformation. If A is not PSD, Rajaeadot remains a bilinear form but is not an inner product.

Computation and variants: If A has structured sparsity or low rank, efficient computation is possible; otherwise it requires O(n^2) operations for two n-vectors. In data analysis, A can encode anisotropic similarity, where directions with larger A-weights contribute more to the measure.

Applications: The concept is used in theoretical discussions of metric learning, generalized similarity measures, and kernel methods, where A represents a learned metric or kernel.

Example: Let x=(1,2), y=(3,4), A=[[2,0],[0,3]]. Then Rajaeadot(x,y)= [1,2] [6,12]^T = 30.

Origin and usage: The term 'rajaeadot' is not standard in textbooks and is presented here as a notational example inspired by the idea of a weighted dot product.