Convex Optimization · Problem 2.18

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · invertible-linear-fractional-functions

Problem

Invertible linear-fractional functions. Let

f : \mathbf{R}^n \to \mathbf{R}^n

be the linear-fractional function

f(x) = (Ax + b)/(c^T x + d), \qquad \mathbf{dom}\, f = \{x \mid c^T x + d > 0\}.

Suppose the matrix

Q = \begin{bmatrix} A & b \\ c^T & d \end{bmatrix}

is nonsingular. Show that

f

is invertible and that

f^{-1}

is a linear-fractional mapping. Give an explicit expression for

f^{-1}

and its domain in terms of

A

b

c

, and

d

. Hint. It may be easier to express

f^{-1}

in terms of

Q