Composition of mixed tensors
= Composition of mixed tensors
{title2=$C^a{}_b=B^a{}_cA^c{}_b$}
A type-$(1,1)$ <tensor> identifies with an <endomorphism> of a finite-dimensional <vector space>. Composing two such maps produces another type-$(1,1)$ <tensor>, because $(X,\eta)\mapsto\eta(B(A(X)))$ is a <multilinear map>. In output-first components the result is $C^a{}_b=B^a{}_cA^c{}_b$; the contracted index implements composition, and a basis change conjugates the corresponding matrices.