Gertrude in Shakespeare's Hamlet, and Queen Elizabeth I have several things in common.
Elizabeth I (daughter of Henry VIII) came to the throne after the death of her half-sister, Mary (also known as "Bloody Mary"). Elizabeth was the next in line. Mary never had children, and upon her death, Elizabeth I became queen because Mary named her heir to the throne.
Gertrude does not rule Denmark, but she comes to the position of Queen honestly. She was married to a revered ruler, Old Hamlet. When he dies, she marries his brother, Claudius. (This is considered an incestuous marriage by Elizabethan standards, but she depends on Claudius to take care of her now.)
Elizabeth is a woman of integrity, with a clear sense of honor to her position as England's monarch; protecting England as a shrewd and intelligent ruler meant everything to her.
When Elizabeth's favorite, Essex, tried to forcibly take the throne from her, Elizabeth was saddened, but her duty to maintain stability in England had to be protected, and she had Essex executed. Mary, Queen of Scots, Elizabeth I's cousin and arch-rival, was a constant threat to the English throne. Although she had been under house- arrest by Elizabeth, it was not until Mary signed a document supporting a plan to kill Elizabeth that the Queen knew she had to stop Mary's plans to take England's throne. It was with regret that Elizabeth executed Mary as well.
Gertrude seems to be a woman of integrity as well. She has foolishly married her brother-in-law. Claudius seems to be fond of Gertrude and exercises some patience initially when Hamlet begins to act crazy. Gertrude loves and worries for Ophelia, has no suspicion about what crimes Claudius has committed, and agrees to the spying on Hamlet in order to help her son.
Queen Elizabeth I was not married. For a long time bringing England out of its weakened state was her only priority. She "flirted" with the monarchs of other global powers (such as the Spanish), but throughout seemed genuinely disinterested in marriage, and upon reaching her forties, her advisors finally dropped the subject. (At this point she needed to name an heir: James VI of Scotland became James I of England when Elizabeth died.)
Claudius and Gertrude have only been married a short time. Hamlet complains that the marriage was so quick that Old Hamlet's funeral food could have been saved for he wedding feast.
...Horatio. The funeral baked meats(185)
Did coldly furnish forth the marriage tables. (II.ii.185-186)
They seem fond of each other, but Claudius is clear where his priorities lie when he allows Gertrude to drink the poisoned wine meant for Hamlet rather than stopping her.
Elizabeth I's reign is one of the longest in England's history.
She ruled alone for nearly half a century...
Claudius reigns only a short time; at the play's beginning, the new King and Queen have only been married a few months. Claudius is not admired. He infers to Laertes that he cannot "punish" Hamlet because he is loved by the people, but Claudius must keep Laertes busy on his return because the people are at the gate calling for Laertes to the their king.
Elizabeth was well-loved. She brought England into a era of peace and prosperity. She brought religious stability; warded off threats by the French and Spanish; rebuilt the treasury; and, brought England into the Elizabethan Renaissance—the rebirth of the arts. Macbeth and his wife were hated and feared. It seems no one wept when they died.