Let ( \ pi, V ) be an irreducible unitary representation of a compact group G . Then every bounded operator T : V \ to V satisfying the property T \ circ \ pi ( s ) = \ pi ( s ) \ circ T for all s \ in G, is a scalar multiple of the identity, i . e . there exists \ lambda \ in \ C such that T = \ lambda \ text { Id }.
42.
Alternatively, observe that two vectors in Euclidean space are linearly independent if and only if one is not a scalar multiple of the other, and that this is clearly the case for " T " ( 1, 0 ) and " T " ( 0, 1 ) ) However, if " T " maps linearly independent vectors to linearly independent vectors, the dimension of its image should be the dimension of its domain, and as such, " T " has rank 2.
43.
More precisely, Feld-Tai reciprocity requires the Hermitian ( or rather, complex-symmetric ) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating \ mathbf { E } and i \ omega \ mathbf { J } is a constant scalar multiple of the operator relating \ mathbf { H } and \ nabla \ times ( \ mathbf { J } / \ varepsilon ), which is true when ? is a constant scalar multiple of ? ( the two operators generally differ by an interchange of ? and ? ).
44.
More precisely, Feld-Tai reciprocity requires the Hermitian ( or rather, complex-symmetric ) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating \ mathbf { E } and i \ omega \ mathbf { J } is a constant scalar multiple of the operator relating \ mathbf { H } and \ nabla \ times ( \ mathbf { J } / \ varepsilon ), which is true when ? is a constant scalar multiple of ? ( the two operators generally differ by an interchange of ? and ? ).
45.
Where ideals are considered equivalent if they are equal up to an overall ( nonzero ) rational scalar multiple . ( Note that this order need not be the full ring of integers, so nonzero ideals need not be invertible . ) Since an order in a number field has only finitely many ideal classes ( even if it is not the maximal order, and we mean here ideals classes for all nonzero ideals, not just the invertible ones ), it follows that there are only finitely many conjugacy classes of matrices over the integers with characteristic polynomial f ( x ).
