I am working with a matrix on a harmonic oscillator problem and the lowest (absolute) frequency $\omega_0$ where the matrix becomes singular is the resonant frequency.
Now I obtained this frequency by calculating it numerically at let's say it is $\omega_0 = 0.3302 - 0.0121i$, ie. a complex frequency.
My question is what is the actual resonant frequency considering there is no such thing as a complex resonant frequency in real life? I know that in various situations you work with complex numbers for convenience and then take the real part at the end to obtain a solution.
However the imaginary part of $\omega_0$ here is necessary to make the matrix singular, if I just take $\omega_0 = 0.3302$ the matrix is not singular and $w_0$ will not be a resonant frequency.
Or have I got it wrong and is it actually fine to just take the real part (and the imaginary part can be sort of considered to be implied)?