The image of a given orthonormal basis for a separable Hilbert space \(\mathcal {H}\) under a bijective, bounded, and linear operator acting on \(\mathcal {H}\) is called a Riesz basis of \(\mathcal {H}\) . Contrary to what happens with Riesz bases (in the usual sense) in separable Hilbert spaces, it is not true in general that the image of a matrix-valued orthonormal basis under a bounded, linear, and bijective operator on \(L^2(G, \mathbb {C}^{s\times r})\) is also a basis and frame for the space \(L^2(G, \mathbb {C}^{s\times r})\) , where G is a \(\sigma \) -compact and metrizable locally compact abelian (LCA) group. We give some classes of operators for the construction of matrix-valued Riesz bases from orthonormal bases of the space \(L^2(G, \mathbb {C}^{s\times r})\) . Motivated by a result due to Holub, we show that a bounded, linear, and bijective operator acting on \(L^2(G, \mathbb {C}^{s\times r})\) which is adjointable with respect to the matrix-valued inner product is positive if and only if it maps a matrix-valued Riesz basis of the space \(L^2(G, \mathbb {C}^{s\times r})\) to its dual Riesz basis.
