S = [ vec2mat(b) for b in W.basis() ]
Qs = []
- for s in basis:
+ for s in S:
# Brute force the multiplication-by-s matrix by looping
# through all elements of the basis and doing the computation
# to find out what the corresponding row should be. BEWARE:
# constructor uses ROW vectors and not COLUMN vectors. That's
# why we're computing rows here and not columns.
Q_rows = []
- for t in basis:
+ for t in S:
this_row = mat2vec((s*t + t*s)/2)
Q_rows.append(W.coordinates(this_row))
- Q = matrix(field,Q_rows)
+ Q = matrix(field, W.dimension(), Q_rows)
Qs.append(Q)
return Qs
+def _embed_complex_matrix(M):
+ """
+ Embed the n-by-n complex matrix ``M`` into the space of real
+ matrices of size 2n-by-2n via the map the sends each entry `z = a +
+ bi` to the block matrix ``[[a,b],[-b,a]]``.
+
+ EXAMPLES::
+
+ sage: F = QuadraticField(-1,'i')
+ sage: x1 = F(4 - 2*i)
+ sage: x2 = F(1 + 2*i)
+ sage: x3 = F(-i)
+ sage: x4 = F(6)
+ sage: M = matrix(F,2,[x1,x2,x3,x4])
+ sage: _embed_complex_matrix(M)
+ [ 4 2| 1 -2]
+ [-2 4| 2 1]
+ [-----+-----]
+ [ 0 1| 6 0]
+ [-1 0| 0 6]
+
+ """
+ n = M.nrows()
+ if M.ncols() != n:
+ raise ArgumentError("the matrix 'M' must be square")
+ field = M.base_ring()
+ blocks = []
+ for z in M.list():
+ a = z.real()
+ b = z.imag()
+ blocks.append(matrix(field, 2, [[a,-b],[b,a]]))
+ return block_matrix(field, n, blocks)
+
+
def RealSymmetricSimpleEJA(n):
"""
The rank-n simple EJA consisting of real symmetric n-by-n