We show that integrated optical volume holographic devices may implement arbitrary linear transformations with order unity efficiency. At the diffraction limit, the device area needed to implement an N × N transformation is proportional to N 2 λ 2 , but in materials with a maximum index modulation of Δ n max , the area scales as N 32 λ 2 Δ n max . This scaling represents a factor of a N improvement over two-port device networks, such as Mach-Zehnder arrays. A holographic system may thus implement 1000 × 1000 transformations in 30 × less area and 10, 000 × 10, 000 transformations in 100 × less area. We explore the practical implications of these scaling laws in the design of electro-optically controlled holographic vector matrix multipliers.