This work studies the problem of reliably estimating a subspace in a federated setting, when some nodes’ outputs can be compromised by Byzantine attacks. Typically, the subspace of interest is the principal subspace of an unknown symmetric matrix. Each node has access to data that can be used to estimate this matrix and its principal subspace. This metaproblem occurs in various applications; two important examples are federated PCA, and the spectral initialization step of iterative solutions to various low-rank (LR) matrix recovery problems in federated settings. We introduce a novel solution framework called Subspace-Median to solve this problem in a provably Byzantine-resilient, and communication-efficient fashion.