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Let's say we have a mathematical expression, $$ \mathbf{y} = \mathbf{Ax}, $$ where $\mathbf{y}$ and $\mathbf{x}$ are a vector, and $\mathbf{A}$ is a matrix. Let's say the vector $\mathbf{y}$ is used to calculate a loss function $\mathcal{L}$ and the gradient $\mathbf{g_y} = \partial\mathcal{L}/\partial\mathbf{y}$ (which is also a vector) is available.

For real-valued tensors, I can understand providing grad_outputs in torch.autograd.grad, e.g. torch.autograd.grad(y, x, grad_outputs=gy) is like calculating the expression below, $$ \frac{\partial\mathcal{L}}{\partial\mathbf{x}} = \mathbf{g_y}^T \frac{\partial\mathbf{y}}{\partial\mathbf{x}}. $$

My question is: what mathematical expression does it calculate if all of the variables are complex?

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