Suppose that a matrix
A = x1B+x2C
is a linear combination of two self-adjoint matrices B and C.
I'm interested in when A represents a physical quantity.
When the linear combination is a complex combination, then B and C have to be commutable for A to represent any physical quantity, cf. this Phys.SE post.
Now suppose that x1 and x2 are real. What happens in this case? If B and C are noncommutable, does A still represent physical quantity?
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