A nonsingular real matrix is said to be perfectly conditioned if its condition number is equal to one. In this paper it is shown that a nonsingular real matrix is perfectly conditioned if and only if the matrix is a nonzero scalar multiple of a norm-preserving linear operator. Therefore the matrices which are perfectly conditioned with respect to the spectral norm are the positive scalar multiples of the orthogonal matrices. The classes of perfectly conditioned matrices are somewhat limited. However they may be of interest to applications in perturbation theory and numerical methods for error bound studies.