# A physical quantity that is a real combination and commutability

Suppose that a matrix

$$A ~=~ x_1 B + x_2 C$$

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 $x_1$ and $x_2$ are real. What happens in this case? If $B$ and $C$ are noncommutable, does $A$ still represent physical quantity?

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Yes, if $B$ and $C$ are self-adjoint $n\times n$ matrices (and even if they are mutually non-commuting), and if $x_1$ and $x_2$ are real numbers, then
$$A= x_1 B + x_2 C$$
is also a self-adjoint matrix $A\!=\!A^{\dagger}$, and hence a physical observable.
In infinite dimensions, $C$ is Hermitian but not necessarily self-adjoint. One needs some domain compatibility condition for the selfadjointness of $C$. –  Arnold Neumaier Sep 27 '12 at 17:33