How does Shakespeare present madness in 'Hamlet'?

Shakespeare leaves the question of Hamlet's madness, the critical crux of the play, as open to interpretation, often dependent on the perspective of those making the judgement.In conformity with his declared intentions, Hamlet's madness is viewed by many as simply a mask or disguise in order to spy on Claudius . This view is shared by those Hamlet confides in, such as Horatio, who has faith Hamlet's sanity throughout. Indeed, this could be entirely due to the perspective of those closest to Hamlet, and his actions are instead branded as 'mad' by those who do not understand them. For example, to be seen as 'mad' could simply mean being the moral minority, as Hamlet declares: 'to be honest, as this world goes, is to be one man picked out of ten thousand'.However, from the perspective of Rosencrantz and Guildenstern, other members of the court and multiple critics, Hamlet's madness appears to be genuine, rather than simply a convincing performance. Many members of the court are shocked by Hamlet's erratic behaviour, such as his rash stabbing of Polonius and his seeming lack of remorse. Indeed, Hamlet appears to alienate those closest to him in this madness, such as Ophelia and Gertrude, perhaps driven to this level of insanity by the orders of the his late father's ghost to revenge his murder.The presentation of Hamlet's madness, therefore, is a complex issue, in which its authenticity appears to depend on the perspective and sympathies of those making the judgement.