The Byzantine Imperial flag is yellow with a black crowned double-headed eagle.
The double-headed eagle was the symbol of the Palaiologos, the last Greek-speaking "Roman" dynasty to rule from Constantinople. Emperor Michael VIII Palaiologos recaptured Constantinople from the Crusaders in 1261, from a state based in Asia Minor; the double-headed eagle symbolized the dynasty's interests in both Asia and Europe, and was kept despite the fact that virtually all of the Asian possessions were gobbled up by theOttomans within a generation of the recapture of the city. Michael's descendants stayed on the Byzantine throne until the city and the Empire fell to the Ottomans in 1453.
The double-headed eagle is much older than Paleologues and Christianity, but in that time it became the symbol of entire Empire. Different colors of eagle had different rank. Some authors said that the gold eagle was reserved for royal family. Silver represented the second rank (despots, sevastokrators - the highest feudal title). Black eagles were used during the war. There again, yellow (gold) was reserved for the Emperor, all other ranks and units had different colors.
Constantinople, the centre of the Byzantine Empire, was the successor of Rome who continued the use of the old imperial single-headed eagle. In Rome, itsymbolized imperial authority and the bronze aquilas and vexilloids (similar to flags) were used whenever the emperor was present. The single-headed eagle turned double when Isaac I Komnenos, sometime from 1057-1059, modified it after being influenced from local traditions from his native Paphlagonia in Asia Minor.
It was also used in the Coat of Arms of the Bulgarian Tsar Ivan Alexander in the 14th century and the Serbian Namanjic dynasty as they too claimed the imperial throne of Constantinople after its fall. George Kastrioti also adopted the double-headed eagle in his flag for the Albanians struggle against the Ottoman Empire. They continue to use it to this day as well as many other nations’ coat of arms, most notably Germany. In Greece, the double-headed eagle is used as the symbol of the Greek Orthodox Church, whose Patriarch still resides in Constantinople. It is also used by the Hellenic Army as their coat of arms.
The double-headed eagle had in the two centuries of Paleologue rule become identified not just with the dynasty but with the Empire itself and, more generally, with institutions and cultural ideas outside the Byzantine Empire that still remained centered on Constantinople.
Most obvious of these is the Greek Orthodox Church, centered in theory in Istanbul to this day, and so it is not surprising that the Church would use the flag. Michael VIII Paleologue adopted this symbol after he had reconquered Constantinople from the Crusaders in 1261. It represented looking towards the East (Asia Minor, traditional power center of the Byzantine-government in exile after the IVth Crusade) and the West (newly re conquered land in Europe).
The double-headed eagle had in the two centuries of Palaiologos rule become identified not just with the dynasty but with the Empire itself and, more generally, with institutions and cultural ideas outside the Byzantine Empire that still remained centered on Constantinople. Most obvious of these is the Greek Orthodox Church, centered in theory in Istanbul to this day, and so it is not surprising that the Church would use the flag.