\(\displaystyle A_{n} \in \mathbb{C} \) is a square hermitian matrix.
\(\displaystyle \alpha \in \mathbb{C} \) is a scalar.
Show a necessary and sufficient condition for the matrix \(\displaystyle \alpha.A \) to be hermitian.
My thought process so far:
\(\displaystyle \alpha = a+ bi \) .
\(\displaystyle (A)_{i,j} = (A^*)_{i,j} = c + di \) .
\(\displaystyle ((\alpha.A)^*)_{i,j} = (\alpha.A)_{i,j} \) .
\(\displaystyle (\alpha.A)_{i,j} = ac + bci + adi - bc \) .
\(\displaystyle ((\alpha.A)^*)_{i,j} = ac - bci - adi - bc \) .
Don't know where to go from here...
\(\displaystyle \alpha \in \mathbb{C} \) is a scalar.
Show a necessary and sufficient condition for the matrix \(\displaystyle \alpha.A \) to be hermitian.
My thought process so far:
\(\displaystyle \alpha = a+ bi \) .
\(\displaystyle (A)_{i,j} = (A^*)_{i,j} = c + di \) .
\(\displaystyle ((\alpha.A)^*)_{i,j} = (\alpha.A)_{i,j} \) .
\(\displaystyle (\alpha.A)_{i,j} = ac + bci + adi - bc \) .
\(\displaystyle ((\alpha.A)^*)_{i,j} = ac - bci - adi - bc \) .
Don't know where to go from here...