stub matching vs quarter wavelength transmission line

quarter wave length transformer is used only for real impedance matching but stubs are used for any case

Stub matching can compensate only immaginary parts of an impedance...so to match the real part you use a lambda/4 transformer and if you need to match even the immaginary part you can use a stub...

a l-long stub has impedance equal to: Z(l) = Zo*[Zc+j Zo tan (bl)]/[Zo+j Zc tan (bl)] where Zc is the impedance of the termination (a open or a short) and Zo is the characteristic impedance of the line...

So a stub based on an open circuit has impedance: Z(l) = -j* Zo cotg (bl).
Alternatively a stub with a short has impedance Z(l) = j Zo tan (bl)

As you can see it is EXCLUSIVELY immaginary , i.e. , it is a pure admittance.

Chosing properly l one can tune the immaginary part to the needed value.

I hope i did no mistake in formulas...

Refer book by
John D. Ryder
Transmission lines & fields.
&
Radio engineering by Terman
This will clear all your doubts regarding matching.