Common quadratic forms

Common quadratic forms

Usage

xTAx(x, A)

xAxT(x, A)

xTAx_solve(x, A, ...)

xTAx_qrsolve(x, A, tol = 1e-07, ...)

sandwich_solve(A, B, ...)

xTAx_eigen(x, A, tol = sqrt(.Machine$double.eps), ...)

sandwich_sginv(A, B, ...)

sandwich_ginv(A, B, ...)

Arguments

x

a vector

A

a square matrix

...

additional arguments to subroutines

tol

tolerance argument passed to the relevant subroutine

B

a square matrix

Details

These are somewhat inspired by emulator::quad.form.inv() and others.

Functions

• xTAx(): Evaluate \(x'Ax\) for vector \(x\) and square matrix \(A\).

• xAxT(): Evaluate \(xAx'\) for vector \(x\) and square matrix \(A\).

• xTAx_solve(): Evaluate \(x'A^{-1}x\) for vector \(x\) and invertible matrix \(A\) using solve().

• xTAx_qrsolve(): Evaluate \(x'A^{-1}x\) for vector \(x\) and matrix \(A\) using QR decomposition and confirming that \(x\) is in the span of \(A\) if \(A\) is singular; returns rank and nullity as attributes just in case subsequent calculations (e.g., hypothesis test degrees of freedom) are affected.

• sandwich_solve(): Evaluate \(A^{-1}B(A')^{-1}\) for \(B\) a square matrix and \(A\) invertible.

• xTAx_eigen(): Evaluate \(x' A^{-1} x\) for vector \(x\) and matrix \(A\) (symmetric, nonnegative-definite) via eigendecomposition and confirming that \(x\) is in the span of \(A\) if \(A\) is singular; returns rank and nullity as attributes just in case subsequent calculations (e.g., hypothesis test degrees of freedom) are affected.

  Decompose \(A = P L P'\) for \(L\) diagonal matrix of eigenvalues and \(P\) orthogonal. Then \(A^{-1} = P L^{-1} P'\).

  Substituting, $$x' A^{-1} x = x' P L^{-1} P' x = h' L^{-1} h$$ for \(h = P' x\).