Evaluate a matrix-induced inner product

Computes \(\langle\boldsymbol{x},\boldsymbol{y}\rangle_{\boldsymbol{A}} =\boldsymbol{x}^{\mathsf{T}}\boldsymbol{A}\boldsymbol{y}\). When \(\boldsymbol{A}\) is symmetric positive definite (as for a non-degenerate Gaussian covariance or precision), this is the \(\boldsymbol{A}\)-inner product on \(\mathbb{R}^{p}\) (https://en.wikipedia.org/wiki/Inner_product_space#Basic_properties). Prefer this name over the generic "bilinear form" when \(\boldsymbol{A}\) is SPD; the Euclidean inner product is the special case \(\boldsymbol{A}=\mathbf{I}\) (\(\boldsymbol{x}^{\mathsf{T}}\boldsymbol{y}\)).

Usage

.inner_product(x, A, y = x)

Arguments

x

Numeric vector.

A

Numeric square matrix, compatible with x and y (SPD when interpreted as an inner-product metric).

y

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

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).

Examples

.inner_product(c(1, 2), diag(2), c(3, 4))
#> [1] 11