# What are the different ways of representing unitary gates in terms of the matrix?

What are the different ways of representing unitary gates in terms of the matrix? In other words what would be the two different matrices which on multiplication give you unitary gate matrix?

A unitary gate is described by a unitary matrix $$U$$. (Note that this gate can have different representations as a matrix depending on what basis you choose to use.)
If you want to decompose that into a pair of matrices that multiply together to give the original unitary, there are infinitely many ways to do this. For example, a pair of matrices $$V$$ and $$V^\dagger U$$ satisfy this for any unitary $$V$$ and, indeed, you can generalise it to the case of $$V$$ simply being invertible. If you’re looking for some particular consistent way of doing it, you might be interested in LU or QR decompositions of a matrix, but since you give no guidance as to the context that you want this for, it’s impossible to know!