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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 (6)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (3)

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}

Applications (3)

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

In[5]:= ConjugateTranspose[{{a, b}, {c, d}}]
Out[5]= {{Conjugate[a], Conjugate[c]}, {Conjugate[b], Conjugate[d]}}

In[6]:= m = {{1, I}, {-I, 2}}; ConjugateTranspose[m] == m
Out[6]= True

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 does not change the shape of vec (matches the Mathematica convention).
  • For symbolic entries, Conjugate[x] is left wrapped around x.
  • Works on higher-rank tensors with the same spec semantics as Transpose.
  • Reads and returns a packed list at every rank, including the rank-1 form. A real buffer skips the conjugation entirely ā€” it is the identity there, and running it would be a second full pass over the data.

Attributes: Protected.

References

See also: Transpose

Notes & additional examples

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.