The Loss
For a target distribution p and model distribution q, cross-entropy is the expected value of -log q under p. In a one-hot classification target, that reduces to the negative log probability assigned to the correct class.
That is why the same training expression is often called negative log likelihood.
Why The Target Matters
The expectation is taken with respect to the target distribution. The model distribution supplies the probabilities being scored, but the target distribution says which outcomes matter for the sample or dataset.
This is the useful intuition: the loss punishes the model when it assigns low probability to outcomes the target says should be likely.
Where Matrix Exponentials Fit
Softmax contains exponentials, and matrix exponentials appear in other mathematical settings, but cross-entropy itself does not require a quantum or eigendecomposition interpretation. The classification case is already useful: logits become probabilities, and the loss scores those probabilities against the target.