How do I construct a controlled Hadamard gate?

[DrHix]

Homework Statement

I am supposed to construct a controlled Hadamard gate
using only single qubit and CNOT gates.

Homework Equations

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We know that any arbitrary unitary Operator U can be written as the Martrix product U=AXBXC, where X is the NOT-Matrix and ABC=1 (identity matrix)
I've already shown that any arbitrary controlled operator can be written as CU=Cphase* (A⊗1)*CNOT*(B⊗1)*CNOT*(C⊗1), with ABC=1

The other relevant equations are the standard equations from quantum mechanics

The Attempt at a Solution

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I've tried everything, LU-Decomposition, QR, Diagonalization etc. I've read something about Sine-Cosine-Decomposition, I think this might be the right direction.

I've also read something about Lie-Groups, though unfortunately, the Hadamard gate is not part of SO(2), only of O(2), there's way more explanation on the generators of SO (2), especially SU (2), and unfortunately I don't know that much about group theory.

 

[Asim Riaz]

I'm not sure if this is the right path and I'm a bit lost on how to continue. Any help would be greatly appreciated.