Equivalence and invariance of the chi and Hoffman constants of a matrix

We show that the following two condition measures of a full column rank matrix $A\in {\mathbb{R}}^{m\times n}$ are identical: the chi constant and a signed Hoffman constant. This identity is naturally suggested by the evident invariance of the chi constant under sign changes of the rows of $A$. We also show that similar equivalence and invariance properties extend to variants of the chi and Hoffman constants that depend only on the linear subspace $A({\mathbb{R}}^n):=\{Ax:x\in {\mathbb{R}}^n\}\subseteq {\mathbb{R}}^m$. Finally, we show similar identities between the chi constants and signed versions of Renegar’s and Grassmannian condition measures.