Evaluate a matrix-induced bilinear form

Computes \(\boldsymbol{x}^{\mathsf{T}}\boldsymbol{A}\boldsymbol{y}\). When \(\boldsymbol{A}\) is symmetric positive definite (as for a non-degenerate Gaussian covariance or precision), this is the \(\boldsymbol{A}\)-inner product. Prefer this name over "dot product", which is reserved for \(\boldsymbol{x}^{\mathsf{T}}\boldsymbol{y}\).

Usage

.bilinear_form(x, A, y = x)

Arguments

x

Numeric vector.

A

Numeric square matrix, compatible with x and y.

y

Numeric vector of the same length as x (default x, which yields the quadratic form \(\boldsymbol{x}^{\mathsf{T}}\boldsymbol{A}\boldsymbol{x}\)).

Value

Numeric scalar.

Details

Implementation uses base::crossprod() as drop(crossprod(x, A %*% y)), which is the standard efficient route to a bilinear / quadratic form in R (avoids an explicit transpose of \(\boldsymbol{x}\) and a temporary outer product).