How is Gertrude in Hamlet similar to Queen Elizabeth I?

Gertrude and Elizabeth both were queens, and both had, as all people in power do, to navigate the corruption that always forms around power. They had to deal with the flattering courtiers that surround a court, hoping to get ahead by pleasing those in power.
Both women also had to deal with patriarchy, or a society run by men. It is here that their differences come to the fore. Gertrude immediately remarried, supporting the dominant ideology that a man should be in charge. She, unfortunately, chose an evil spouse, one who had, in fact, murdered her first husband. However, she might have decided she would have more power as a wife than as a mother to a king; one imagines that her son Hamlet would have taken over had she not remarried.
Elizabeth, on the other hand, skillfully maneuvered around the issue of marriage, using herself as bait and pretending she was interested in marrying this or that European royal. This kept her enemies in check while, at the same time, she avoided marriage and the loss of power that would have caused her.