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

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 $$A^*=A^t$$