no matter vs irrespective

no matter and irrespective both are adverbs.

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

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

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

Other synonyms of no matter include disregarding, disregardless, regardless.

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