Transpose vs Conjugate Transpose
Transpose of a matrix A can be identified as the matrix obtained by rearranging the columns as rows or rows as columns. As a result, each element’s indices are interchanged. More formally, transpose of a matrix A, is defined as
where
In a transpose matrix, the diagonal remains unchanged. But all the other elements are rotated around the diagonal. Also, the size of the matrices also changes from m×n to n×m.
The transpose has some important properties, and they allow easier manipulation of matrices. Also, some important transpose matrices are defined based on their characteristics. If the matrix is equal to its transpose, then the matrix is symmetric. If the matrix is equal to its negative of the transpose, then the matrix is a skew symmetric.
The conjugate transpose of a matrix is the transpose of the matrix with the elements replaced with its complex conjugate. That is, the complex conjugate (A*) is defined as the transpose of the complex conjugate of matrix A.
A*=(Ā )T; In detail,
where
and āji ε C.
It is also known as the Hermitian transpose and Hermitian conjugate. If the conjugate transpose is equal to the matrix itself, the matrix is known as a Hermitian matrix. If conjugate transpose is equal to the negative of the matrix, it is a skew Hermitian matrix. And if the inverse of the matrix is equal to the complex conjugate, the matrix is unitary.
Likewise, all the special matrices complex conjugate also has special properties that can be used to mathematically manipulate them easily. The conjugate transpose is widely used in the quantum mechanics and its relevant fields.
What is the difference between Transpose and Conjugate Transpose?
• Transpose of a matrix is obtained by rearranging columns into rows, or rows into columns. The complex conjugate of a matrix is obtained by replacing each element by its complex conjugate (i.e x+iy ⇛ x-iy or vice versa). The conjugate transpose is obtained by performing both operations on the matrix.
• Therefore, conjugate transpose is just a transpose matrix with its complex conjugates as the elements.