Transpose/Conjugate Transpose of a Matrix

1. The Transpose of any Matrix can be obtained by changing the Rows of Matrix to Columns and Columns of Matrix to Rows.
2. The Transpose of an \(M \times N\) Matrix is an \(N \times M\) Matrix.
3. The following example gives a \(3 \times 2\) Matrix \(A\) and its Transpose \(2 \times 3\) Matrix \(A^T\)
   
   \(A=\begin{bmatrix} 3 & 7\\ 2 & 9 \\4 & 1 \end{bmatrix}\hspace{.8cm}A^T=\begin{bmatrix} 3 & 2 & 4\\ 7 & 9 & 1 \end{bmatrix}\)
4. The Conjugate Matrix of any Complex Matrix can be obtained by Converting all Complex/Imaginary Elements of the Matrix into their Corresponding Conjugate Values.
5. The Conjugate Transpose of any Complex Matrix can be obtained by Transposing the Conjugate of the Matrix or Conjugating the Transpose of the Matrix.
6. The following example gives a \(3 \times 2\) Complex Matrix \(A\) and its Conjugate Transpose \(2 \times 3\) Matrix \(A^\dagger\)
   
   \(A=\begin{bmatrix} 3 +8i & 7-2i\\ -2-4i & 9+i \\i & 1-5i \end{bmatrix}\hspace{.8cm}A^\dagger=\begin{bmatrix} 3-8i & -2+4i & -i\\ 7+2i & 9-i & 1+5i \end{bmatrix}\)