Lesson 6 Transposition and conjugation

The transpose 

, conjugate 

,
and adjoint (i.e., conjugate transpose) 

 of a matrix or vector 

 are
obtained by the member functions transpose(), conjugate(),
and adjoint(), respectively.

Example:	Output:
MatrixXcf a = MatrixXcf::Random(2,2); cout << "Here is the matrix a\n" << a << endl; cout << "Here is the matrix a^T\n" << a.transpose() << endl; cout << "Here is the conjugate of a\n" << a.conjugate() << endl; cout << "Here is the matrix a^*\n" << a.adjoint() << endl;	Here is the matrix a (-0.211,0.68) (-0.605,0.823) (0.597,0.566) (0.536,-0.33) Here is the matrix a^T (-0.211,0.68) (0.597,0.566) (-0.605,0.823) (0.536,-0.33) Here is the conjugate of a (-0.211,-0.68) (-0.605,-0.823) (0.597,-0.566) (0.536,0.33) Here is the matrix a^* (-0.211,-0.68) (0.597,-0.566) (-0.605,-0.823) (0.536,0.33)

For real matrices, 

conjugate()

 is a no-operation, and so 

adjoint()

 is equivalent to 

transpose()

.

As for basic arithmetic operators, 

transpose()

 and 

adjoint()

 simply return a proxy object without doing the actual transposition. If you do 

b = a.transpose()

, then the transpose is evaluated at the same time as the result
is written into 

b

. However, there is a complication here. If you do 

a = a.transpose()

, then Eigen starts
writing the result into 

a

before the evaluation of the transpose is finished. Therefore, the instruction 

a = a.transpose()

 does not replace 

a

 with its transpose, as one would expect:

Example:	Output:
Matrix2i a; a << 1, 2, 3, 4; cout << "Here is the matrix a:\n" << a << endl; a = a.transpose(); // !!! do NOT do this !!! cout << "and the result of the aliasing effect:\n" << a << endl;	Here is the matrix a: 1 2 3 4 and the result of the aliasing effect: 1 2 2 4

This is the so-called aliasing issue. In "debug mode", i.e., when assertions have
not been disabled, such common pitfalls are automatically detected.

For in-place transposition, as for instance in 

a = a.transpose()

, simply use the transposeInPlace() function:

Example:	Output:
MatrixXf a(2,3); a << 1, 2, 3, 4, 5, 6; cout << "Here is the initial matrix a:\n" << a << endl; a.transposeInPlace(); cout << "and after being transposed:\n" << a << endl;	Here is the initial matrix a: 1 2 3 4 5 6 and after being transposed: 1 4 2 5 3 6

There is also the adjointInPlace() function for complex
matrices.

The word adjoint has a number of related meanings. In linear algebra, it refers to the conjugate transpose and is most commonly
denoted