v1 ιδων (ειδον) "when he saw [the crowds]" - adverbial, probably temporal, forming a causal clause; "Jesus went up the mountain because he saw the crowds.
το ορος "a mountainside" - the definite article may indicate a particular hill, but most likely prompting us to think of Mount Sinai.
καθισαντος (καθιζω) aor. part. "sat down" participle is adverbial, temporal; "when he sat down his disciples came to him." A teacher sits down to teach.
οι μαθηται "the disciples" - Given the context, the beatitudes are for the disciples, but are the disciples the "poor in spirit", etc.? Some scholars take the view that the beatitudes are statements of grace directed to the disciples, "blessed are you, the poor in spirit ...." but the Greek is not that specific (although v11-12 does move to "you", meaning "you disciples.")
v2 ανοιξας το στομα aor. part. “opening [the mouth] attendant circumstance participle expressing action "he opened his mouth...
εδιδασκεν (διδασκω) imperf. "to teach" the imperfect may be used here to make the point that the sermon on the mount is a summary of Jesus' teachings, so "this is what he used to teach".
v3 μακαριοι adj. "blessed" - not the best translation, "fortunate" or "well off", possibly "happy are you". Jesus is telling his disciples that they are fortunate to be this way, fortunate to possess these qualities of life, because in possessing them they inherit God's promised kingdom. The beatitudes are not imperatives so not earned but gifts.
οι πτωχοι adj. "the poor ones" - those in total poverty, possessing nothing and with no means to earn a living other than by receiving alms.
τω πνευματι (α ατος) dat. "in spirit" - dative is local, serves to define the poverty. To be poor in spirit is to be totally destitute spiritually and so recognize the need for a total dependence on God. As Matthew uses this phrase it has nothing to do with actual poverty.
οτι "for" - introducing a causal clause explaining why the poor in spirit are blessed. Note repeated use of this conjunction throughout the beatitudes.
αυτων εστιν "theirs is" – in the sense of consequence (not reward).
των ουρανων (ος) gen. "[the kingdom] of heaven" - the eschatological reign of God.
v4 οι πενθουντες (πενθεω) part. "the mourning ones - participle serves as a substantive. It is likely that the mourning is over sin; "fortunate are those who are broken before God."
παρακληθησονται (παρακαλεω) fut. pas. "will be comforted" - a divine passive; "God will comfort them".
v5 οι πραεις (πραυς) adj. "the gentle ones”. a substantive. "Gentle" in the sense of not demanding of God, so "submissive to the will of God" and therefore willing to look to him for vindication.
κληρονομησουσιν την γην "[for they] will inherit the earth" - lit. receive by lot, therefore "possess" - identifying the consequence of covenant inclusion through submission to the divine will (cf. Psalm 37:11, inheriting the promised land. "They will receive what God has promised").
v6 οι πεινωντες (πειναω) pres. part. "the hungering ones" - a substantive.
την δικαιοσυνην (η) "righteousness" - righteousness, justice. Jesus is not speaking of for social justice, nor even those who desire social justice, but a desire for personal vindication, of being set-right before God, of being "judged in the right" - the same sense of justification expounded by Paul.
χορτασθησονται (χορταζω) fut. pas. "will be satisfied ".
v7 οι ελεημονες (ων ονος) "the merciful ones" – the reciprocal nature of mercy and forgiveness is stressed in the New and Old Testaments. The Lord's Prayer gives the classic example, "forgive us our sins as we forgive those who sin against us." This beatitude is not saying that those who show mercy will have mercy shown to them, ie. a salvation by works idea. Rather that they who show mercy will know mercy.
v8 τη καρδια (α) dat. "[the pure] in heart" - dative of locality, the purity of heart referred to here may be the righteousness of God or the regenerate nature – it is certainly not moral or sexual purity. Those who possess righteousness will "be like him" and …
οψονται (οραω) fut. "will see [God]" – i.e. find themselves in God's presence.
v9 οι ειρηνοποιοι (ος) "the ones who make peace" – a hapax legomenon. Clearly, Jesus is not thinking of conflict-resolution managers. In the end Jesus maybe referring to those who are no longer at war with God.
θεου (ος) gen. "[sons] of God" - genitive is relational; so “members of God's family”, i.e. the baptized.
v10 οι δεδιωγμενοι (διωκω) perf. pas. part. "the persecuted ones" - the perfect tense indicates persecution that began in time past, the consequences of which continue into the present, although with participles, duration is also intended those who endure persecution. The meaning of the word is usually in the NT, "to put to flight" or "drive away", but also carries a positive sense, "to follow with intensity and purpose and effort, in order to catch up with, for friendly or hostile purpose - to run after, to chase after, to pursue. to hasten, to run, to press forward, to press on, to follow without hostile intent". This meaning certainly fits with the other beatitudes; "blessed are you who pursue the living God." The trouble is v10 and 11 indicate the sense is "blessed are you when people persecute you because of your standing with God." So perhaps Jesus is warning his disciples that a servant is not greater than his master; if they persecute the master they will persecute the servant.
ενεκεν + gen. "because of" - causal.
η βασιλεια των ουρανων "the kingdom of heaven" - Matthew ends with a typical inclusio, rounding of the beatitudes as a package by ending where he started.
v11 εστε "are you" - The change from the third person to the second person is important, as noted above. This application of the final beatitude guides us in the application of the other beatitudes.
οταν + subj. "when" –an indefinite temporal clause expressing general time.
ονειδισωσιν (ονειδιζω) aor. subj. "people insult (reproach, upbraid)
ψευδομενοι (ψευδομαι) part. "falsely" - the act of being publically attacked alone does not carry any implied lessing, it iw shen in telling the truth you are maligned falsely, when the 8th commandment is ignored, that God blesses the truth teller.
ενεκεν + gen. "because of [me]" – causal – probably because of their testimony of Christ, but even more likely etter, because of their identification with Christ. This is a foreshadowing of the Passion events, of denial and abandonment by the disciples. They will not be ‘happy’ about the results of those actions.
v12 αγαλλιασθε (αγαλλιαω) imp. "be glad" - exceeding glad.
οτι "because" - a causal clause explaining why the disciples should rejoice and be glad.
ο μισθος "reward" - the reward is divine approval. "payment/wage" is another possible translation but seems crass, this serves to highlight the transitory nature of suffering.
γαρ "for" - Luke uses γαρ for οτι above giving both an equal causal standing. Here the choice of γαρ is causal, “..indeed in just the same way they made the prophets before you suffer persecution...".
ουτως adv. "in the same way" – an idiom, the persecuted believer stands in good company.