Conjugate transpose of a matrix ‘P’ is basically a matrix which is equal to the conjugate of the matrix obtained by taking the transpose of the matrix ‘P’. In order to find the conjugate transpose of any matrix; firstly, transpose is obtained and secondly, the conjugate is obtained. The conjugate transpose is generally denoted as
Conjugate transpose of a matrix example
Let P is a matrix such that
Now, to find the conjugate transpose of this matrix P, we first find the transpose of matrix P i.e.
In the second step, we find conjugate of the matrix PT
This is the conjugate transpose of a 2 x 2 matrix P.
Conjugate transpose of a matrix properties
The conjugate transpose of matrices S and R are Sθ and Rθ, respectively. Then,
1. (Sθ)θ = S and (Rθ)θ = R
2. (S + R)θ = Sθ + Rθ
where ‘a‘ is a complex number.
4. (SR)θ = RθSθ