- #1
MisterX
- 764
- 71
Suppose ##A## is a linear operator ##V\to V## and ##\mathbf{x} \in V##. We define a non-linear operator ##\langle A \rangle## as $$\langle A \rangle\mathbf{x} := <\mathbf{x}, A\mathbf{x}>\mathbf{x}$$
Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B = B\langle A \rangle ## ?
More generally if we have ##Q\mathbf{x} = q(\mathbf{x}) \mathbf{x}## with scalar function ##q##, when does ##AQ=QA ## ? I assert we can choose ##A, Q## such that this is false.
Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B = B\langle A \rangle ## ?
More generally if we have ##Q\mathbf{x} = q(\mathbf{x}) \mathbf{x}## with scalar function ##q##, when does ##AQ=QA ## ? I assert we can choose ##A, Q## such that this is false.