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In Chapter II of Dirac's book Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/eigenvectors exist and (if they do) how to find them.

He then goes on to state that a special tractable case can be found in the event that a real linear operator, $\xi$ can be expressed as the algebraic expression:

$\phi$($\xi$) $\equiv$ $\xi$n + a1 $\xi$n-1 + a2 $\xi$n-2 + ... + an = 0

Where the ai are all numbers. He then factorises this as:

$\phi$($\xi$) $\equiv$ ($\xi$- c1)($\xi$- c2)($\xi$- c3)...($\xi$- cn).

Where the ci are also numbers.

I'm unsure how this factorisation has been achieved. Since Dirac didn't explicitly state the difference between $\phi$($\xi$) and plain $\xi$, then I imagine that $\xi$ represents a given dynamical variable and $\phi$($\xi$) the corresponding operator. This may be incorrect however. This is the only step in the logic that I can't work out, I understand his subsequent arguments but this step is eluding me.

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    $\begingroup$ He factorizes suposing that, in the case that you act over a proper eigenvector, the operator behaves as a normal number, and since you have a polinomial equation, with real values, one can, thus, use the fundamental theorem of algebra do factorize it. $\endgroup$ – Hydro Guy Jul 23 '14 at 14:10
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From the equations, $\phi$ is the operator acting on the variable/state $\xi$. It is important to notice also that in the factorization the $a_i$ numbers are real and the $c_i$ are complex. This factorization comes just from the mathematical fact that for a polynomial equation of degree $n$, such as the first equation, there are $n$ complex roots.

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