Fix a unit vector . Let be another unit vector, orthogonal to both and . Let x and y be an arbitrary pair of unit vectors orthogonal to each other. Then there exists a unitary U with and . Therefore Similarly,

Thus Ax is orthogonal to everything in the orthogonal complement of x, so that Ax must be a multiple of x. What's more, that multiple must be the same for x as it is for . Conclusion: A is a scalar multiple of the identity operator.