Adjoint of Complex Matrix

Question

Assume A is a matrix with complex entries. Prove that all diagonal elements of the matrix (A*)A are real where A* is the adjoint of A.

Where do I start here?

Thanks in advanced!

Answer 1

Let $A \in \mathbb{C}^{m \times n}$ have matrix elements $(A_{ij})$ where $i$ denotes row position and $j$ denotes column position. Then we have that the adjoint $A^* \in \mathbb{C}^{n \times m}$ has elements $(\bar A_{ji})$, with $\bar A_{ji}$ denoting the complex conjugate of $A_{ji}$, and $i$ again denoting row position and $j$ column position.

So from the definition of matrix multiplication, we have the formula for a matrix element $x_{ij}$ of $A^*A$:

$$x_{ij} = \sum_{k=1}^m \bar A_{ki} A_{kj}$$

When we calculate a diagonal element $x_{ii}$ with this formula, we obtain

$$x_{ii} = \sum_{k=1}^m \bar A_{ki} A_{ki} = \sum_{k=1}^m |A_{ki}|^2$$