Section 6.3 Transpose and Conjugate
The transpose of a matrix is obtained by flipping it over its diagonal, swapping rows with columns. The transpose is denoted as \(A^T\) and can be computed in Sage using the
transpose method:
A matrix \(A\) is symmetric if it is equal to its transpose, i.e., \(A = A^T\text{.}\) Sage offers
is_symmetric() method to check if a matrix is symmetric.
In some fields like signal processing and quantum mechanics, working over complex vector spaces is necessary and common. The conjugate transpose matrix is defined as the transpose of the conjugate matrix. In Sage, the conjugate transpose of a matrix is obtained by first computing its conjugate then taking the transpose.
To find the conjugate of a matrix in Sage, we use the
conjugate() method.
Then, we compute the transpose of that conjugate matrix using the transpose method shown earlier.
We can also calculate the conjugate transpose directly using the
conjugate_transpose method.
The conjugate transpose of a matrix \(A\) is called the Hermitian transpose, Hermititan conjugate, or transjugate and is denoted \(A^*\text{,}\) \(A^{H}\) or \(A^{\dagger}\text{.}\)
The conjugate transpose matrix is also called the Adjoint matrix. Historically, this term was also used to refer to Adjugate concept introduced in chapter \(8\text{.}\) To avoid confusion, this terminology will not be used in this book. See Section 8.5 for more details.
