in the context of real value matrix are conjugate transposing and regular transposing the same?

Question

assume a matrix does not have any complex entry.

a = np.matrix(np.arange(6).reshape(3,2))
a
matrix([[0, 1],
        [2, 3],
        [4, 5]])

(regular) transposing this matrix

a.T
matrix([[0, 2, 4],
        [1, 3, 5]])

and conjugate transposing this matrix

a.getH()
matrix([[0, 2, 4],
        [1, 3, 5]])

seems to have the same output.

the conjecture above is based on the Python NumPy.

is it true mathematically?

Answer 1

Given $A\in \textsf{M}_{m\times n}(F)$, their conjugate transpose $A^*\in \textsf{M}_{n\times m}(F)$ is defined as $$(A^*)_{ij}=\overline{A_{ji}}$$ for $1\le i\le n$, $1\le j\le m$. If $F=\mathbb{R}$, then $\overline{A_{ji}}=A_{ji}$. So, yes, it's correct to say that $$A^*=A^t$$