Fault detection and process monitoring using principal-component analysis (PCA) and partial least squares were studied intensively and applied to industrial processes. The fundamental issues of detectability, reconstructability, and isolatability for multidimensional faults are studied. PCA is used to define an orthogonal partition of the measurement space into two orthogonal subspaces, a principal-component subspace, and a residual subspace. Each multidimensional fault is also described by a subspace on which the fault displacement occurs. Fault reconstruction leads to fault identification and consists of finding a new vector in the fault subspace with minimum distance to the principal-component subspace. The unreconstructed variance is proposed to measure the reliability of the reconstruction procedure and determine the PCA model for best reconstruction. Based on the fault subspace, fault magnitude, and the squared prediction error, necessary and sufficient conditions are provided to determine if the faults are detectable, reconstructable, and isolatable.