Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: M = random_matrix(ZZ, 2)
sage: M == M.conjugate()
True
sage: M = random_matrix(QQ, 3)
sage: M == M.conjugate()
True
sage: M = random_matrix(RR, 2)
sage: M == M.conjugate()
True
conjugate_transpose()
Returns the transpose of self after each entry has been converted to its complex conjugate.
Note: This function is sometimes known as the “adjoint” of a matrix, though there is substantial variation
and some confusion with the use of that term.
OUTPUT:
A matrix formed by taking the complex conjugate of every entry of self and then transposing the resulting
matrix.
Complex conjugation is implemented for many subfields of the complex numbers. See the examples below,
or more at conjugate().
EXAMPLES:
sage: M = matrix(SR, 2, 2, [[2-I, 3+4*I], [9-6*I, 5*I]])
sage: M.base_ring()
Symbolic Ring
sage: M.conjugate_transpose()
[ I + 2 6*I + 9]
[-4*I + 3 -5*I]
sage: P = matrix(CC, 3, 2, [0.95-0.63*I, 0.84+0.13*I, 0.94+0.23*I, 0.23+0.59*I, 0.52-0.41*I,
sage: P.base_ring()
Complex Field with 53 bits of precision
sage: P.conjugate_transpose()
[ 0.950... + 0.630...*I 0.940... - 0.230...*I 0.520... + 0.410...*I]
[ 0.840... - 0.130...*I 0.230... - 0.590...*I -0.500... - 0.900...*I]
There is also a shortcut for the conjugate transpose, or “Hermitian transpose”:
sage: M.H
[ I + 2 6*I + 9]
[-4*I + 3 -5*I]
Matrices over base rings that can be embedded in the real numbers will behave as expected.
sage: P = random_matrix(QQ, 3, 4)
sage: P.conjugate_transpose() == P.transpose()
True
The conjugate of a matrix is formed by taking conjugates of all the entries. Some specialized subfields of
the complex numbers are implemented in Sage and complex conjugation can be applied. (Matrices over
quadratic number fields are another class of examples.)
sage: C = CyclotomicField(5)
sage: a = C.gen(); a
zeta5
143