+
+class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
+ r"""
+ The algebra of ``n``-by-``n`` matrices with quaternion entries over
+ (a subfield of) the real numbers.
+
+ The usual matrix spaces in SageMath don't support quaternion entries
+ because they assume that the entries of the matrix come from a
+ commutative ring, and the quaternions are not commutative.
+
+ SETUP::
+
+ sage: from mjo.octonions import QuaternionMatrixAlgebra
+
+ EXAMPLES::
+
+ sage: QuaternionMatrixAlgebra(3)
+ Module of 3 by 3 matrices with entries in Quaternion
+ Algebra (-1, -1) with base ring Algebraic Real Field
+ over the scalar ring Algebraic Real Field
+ sage: QuaternionMatrixAlgebra(3,QQ)
+ Module of 3 by 3 matrices with entries in Quaternion
+ Algebra (-1, -1) with base ring Rational Field over
+ the scalar ring Rational Field
+
+ ::
+
+ sage: A = QuaternionMatrixAlgebra(2)
+ sage: i,j,k = A.entry_algebra().gens()
+ sage: A([ [1+i, j-2],
+ ....: [k, k+j] ])
+ +-------+--------+
+ | 1 + i | -2 + j |
+ +-------+--------+
+ | k | j + k |
+ +-------+--------+
+
+ ::
+
+ sage: A1 = QuaternionMatrixAlgebra(1,QQ)
+ sage: A2 = QuaternionMatrixAlgebra(2,QQ)
+ sage: cartesian_product([A1,A2])
+ Module of 1 by 1 matrices with entries in Quaternion Algebra
+ (-1, -1) with base ring Rational Field over the scalar ring
+ Rational Field (+) Module of 2 by 2 matrices with entries in
+ Quaternion Algebra (-1, -1) with base ring Rational Field over
+ the scalar ring Rational Field
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
+ sage: x = A.random_element()
+ sage: x*A.one() == x and A.one()*x == x
+ True
+
+ """
+ def __init__(self, n, scalars=AA, **kwargs):
+ # The -1,-1 gives us the "usual" definition of quaternion
+ Q = QuaternionAlgebra(scalars,-1,-1)
+ super().__init__(Q, scalars, n, **kwargs)