Consider a vector $\mathbf{x} \in \mathbb{C}$, and its complex conjugate transpose $\mathbf{x}^H$. Computing the outer product of both vectors results the matrix $\mathbf{\Phi} = \mathbf{x} \mathbf{x}^H$.
Now I would like to subtract$\mathbf{\Phi}$ from the identity matrix multiplied with the trace $\text{tr}(\cdot)$ of $\mathbf{\Phi}$:
$\mathbf{\Psi} = \text{tr}(\mathbf{\Phi})\mathbf{I} - \mathbf{\Phi} $.
For my application, it would be preferable if an explicit computation of $\mathbf{\Phi}$ could be avoided.
The trace can be expressed in terms of $\mathbf{x}$ (I think):
$ \text{tr}(\mathbf{\Phi}) = \sum_i \mathbf{x}_i^2$.
My questions are
- Is my reformulation for the trace correct?
- Can $\mathbf{\Psi}$ be expressed in a way that does not require me to compute $\mathbf{\Phi}$?
I am also interested in any rules or concepts that cover or have a name for this specific problem, as I do not have a strong math background.