Adjoint Operator
Table of Contents
Note: The Adjoint Matrix is a different thing. Its the transpose of the Cofactor Matrix. Since 'Adjoint' is more frequently used for Ajoint Operator, it is better to call it 'Adjugate Matrix' instead of 'Adjoint Matrix'.
Each linear operator \(A\) on an Inner Product Space defines a Hermitian adjoint operator \(A^*\) such that:
\begin{equation*} \langle Ax, y \rangle = \langle x, A^* y \rangle \end{equation*}Where \(\langle \cdot , \cdot \rangle\) is the inner product.
Adjoint operator is also called:
- Adjoint
- Hermitian Conjugate
- Hermitian
For real matrices, Transpose is the adjoint.
For complex matrices, Complex conjugate of the transpose is the adjoint.
1. Self Adjoint Operator
An operator whose adjoint operator is itself. i.e.
\begin{equation*} \langle Ax, y \rangle = \langle x, Ay \rangle \end{equation*}- Symmetric Matrix are Self Adjoint
- Hermitian Matrix are Self Adjoint
2. Complex Conjugate for Complex Matrices
https://x.com/bpanthi977/status/1846632664595312737
Why do we take complex conjugate of transpose for Complex matrices in places where just transpose works for Real matrices?
Because, inner product (dot product) for complex vectors needs to be defined as <x, y> = ∑ xi yi* to satisfy <x,x> ≥ 0 condition on inner product
Transpose and conjugate transpose are just different ways of finding the `Adjoint` for different kinds of matrices (real, complex).
where Adjoint of A is such an A* that satisfies:
<Ax, y> = <x, A*y>
