# openfermion.linalg.givens_decomposition

Decompose a matrix into a sequence of Givens rotations.

The input is an $$m \times n$$ matrix $$Q$$ with $$m \leq n$$. The rows of $$Q$$ are orthonormal. $$Q$$ can be decomposed as follows:

$V Q U^\dagger = D$

where $$V$$ and $$U$$ are unitary matrices, and $$D$$ is an $$m \times n$$ matrix with the first $$m$$ columns forming a diagonal matrix and the rest of the columns being zero. 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_rows A numpy array or matrix with orthonormal rows, representing the matrix Q.

## Returns

givens_rotations (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\$$.
left_unitary (ndarray):
An \$$m \times m\$$ numpy array representing the matrix
\$$V\$$.
diagonal (ndarray):
A list of the nonzero entries of \$$D\$$.

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