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ConjugateTranspose

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

ConjugateTranspose[m]
    Gives the conjugate transpose of m, equivalent to Conjugate[Transpose[m]].
ConjugateTranspose[m, spec]
    Gives Conjugate[Transpose[m, spec]], permuting the levels of m according to
    the spec list and then conjugating every entry.
    On a 1-D vector, ConjugateTranspose[vec] conjugates the entries without
    changing the shape of vec.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= ConjugateTranspose[{{1, 2 I, 3}, {3 + 4 I, 5, I}}]
Out[1]= {{1, 3 - 4*I}, {-2*I, 5}, {3, -I}}

In[2]:= ConjugateTranspose[{{a + b I, c + d I}}]
Out[2]= {{Conjugate[a + I b]}, {Conjugate[c + I d]}}

In[3]:= ConjugateTranspose[{1, 2 I, 3 + 4 I}]
Out[3]= {1, -2*I, 3 - 4*I}

Implementation notes

builtin_conjugate_transpose (in src/list.c) is a thin composition over existing primitives. It first checks the argument is a rectangular nested List via get_array_dimensions; a symbolic (non-list) matrix is left unevaluated so ConjugateTranspose[A] survives. For a 1-D vector it just maps Conjugate elementwise. Otherwise it builds and evaluates Transpose[m] (or Transpose[m, spec]), then conjugates the transposed result. All heavy lifting is delegated to Transpose and Conjugate through eval_and_free.

  • Protected.
  • On a 1-D vector, ConjugateTranspose[vec] conjugates the entries but

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= ConjugateTranspose[{{1 + I, 2}, {3, 4 - I}}]
Out[1]= {{1 - I, 3}, {2, 4 + I}}

In[1]:= ConjugateTranspose[{{a, b}, {c, d}}]
Out[1]= {{Conjugate[a], Conjugate[c]}, {Conjugate[b], Conjugate[d]}}

In[1]:= m = {{1, I}, {-I, 2}}; ConjugateTranspose[m] == m
Out[1]= True

Notes

ConjugateTranspose[m] is the Hermitian adjoint Conjugate[Transpose[m]]. The last example confirms a matrix is Hermitian (equal to its own conjugate transpose). On a vector it conjugates the entries in place.