# openfermion.linalg.givens_decomposition_square

Decompose a square matrix into a sequence of Givens rotations.

### Used in the notebooks

Used in the tutorials

The input is a square $n \times n$ matrix $Q$. $Q$ can be decomposed as follows:

$$Q = DU$$

where $U$ is unitary and $D$ is diagonal. Furthermore, we can decompose $U$ as

$$U = G_k ... G_1$$

where $G_1, \ldots, G_k$ are complex Givens rotations. A Givens rotation is a rotation within the two-dimensional subspace spanned by two coordinate axes. Within the two relevant coordinate axes, a Givens rotation has the form

$$\begin{pmatrix} \cos(\theta) & -e^{i \varphi} \sin(\theta) \\ \sin(\theta) & e^{i \varphi} \cos(\theta) \end{pmatrix}.$$

unitary_matrix A numpy array with orthonormal rows, representing the matrix Q.

## Returns

decomposition (list[tuple]):
A list of tuples of objects describing Givens
rotations. The list looks like [(G_1, ), (G_2, G_3), ... ].
The Givens rotations within a tuple can be implemented in parallel.
The description of a Givens rotation is itself a tuple of the
form $(i, j, \theta, \varphi)$, which represents a
Givens rotation of coordinates
$i$ and $j$ by angles $\theta$ and
$\varphi$.
diagonal (ndarray):
A list of the nonzero entries of $D$.

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