irrespective vs no matter

irrespective and no matter both are adverbs.

Word	Noun	Adjective	Verb	Adverb
irrespective	No	No	No	Yes
no matter	No	No	No	Yes

As adverbs, irrespective and no matter are synonyms defined as:

irrespective and no matter: in spite of everything; without regard to drawbacks

Other synonyms of irrespective include disregarding, disregardless, regardless.

irrespective (adverb)	no matter (adverb)
in spite of everything; without regard to drawbacks	in spite of everything; without regard to drawbacks