WeightedGramMatrix. Consider a multi-objective least squares problem with matrices A1,...,Ak and positive weights 1,...,k.

The matrix G=1A1TA1+...+kAkTAk is called the weightedGrammatrix; it is the Gram matrix of the stacked matrix A~ given by

1A1

.

A~= .

.

kAk

associated with the multi-objective problem.

Show that G is invertible provided there is no nonzero vector x that satisfies A1x=0,...,Akx=0.