Motivation for Conjugate transpose of a matrix

Question

I'am currently going through a self study of Linear algebra . I'am finding it difficult to grasp the intuition behind the concept of Conjugate transpose of a matrix .Why take the complex conjugate of each entry after a transpose for a complex matrix ?Can someone help me fill that gap ?

Answer 1

The idea behind this in the end is that a useful scalar product on $\mathbb C^n$ is given by $\langle x, y\rangle=\sum_{k=1}^n \overline {x_k}y_k$ (and not $\sum_{k=1}^n {x_k}y_k$ as in the real case).

Given any complex $n\times n$ matrix $A$ we can consider the bilinear map $$\begin{align}f_A\colon \mathbb C^n\times \mathbb C^n&\to\mathbb C\\(x,y)&\mapsto\langle x,Ay\rangle\end{align}$$ which has some interesting properties. Just as well we might have considered for a complex $n\times n$ matrix $B$ the map $$\begin{align}g_B\colon \mathbb C^n\times \mathbb C^n&\to\mathbb C\\(x,y)&\mapsto\langle Bx,y\rangle\end{align}$$ In the end it doesn't matter if we onsider the first or second kind of maps (or even a combination thereof)! This is because for each $A$ there exists a unique $B$ such that $g_B=f_A$ (and vice versa). And guess what? $B=\overline {A^T}$.