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This video explains that the cross product and the determinant involve the dot product under the hood.

This video explains that the most fundamental, entry-wise perspective of matrix multiplication involves the dot product under the hood.

When working with inner products other than the dot product, should/can cross products, determinants, and matrix multiplication be computed in non-standard ways, built from the particular inner product under consideration?

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  • $\begingroup$ There's no such thing as "the" dot product; you can just change the basis to make any other inner product you had in mind look like you expect a dot product to look. $\endgroup$
    – J.G.
    Dec 14, 2018 at 18:26

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For a general finite-dimensional Hilbert space (with general inner product), you can formulate an analog between matrix multiplication (operators in Euclidean space) and an array of linear combinations of basis vectors in your Hilbert space, $H$. For example, matrix-vector multiplication in a general finite-dimensional Hilbert space is equivalent to a finite linear combination of basis functions in $H$. The analog of Matrix-matrix multiplication in $H$ would correspond to an array of different finite linear combinations of the same set of basis function in $H$.

But I'm not so sure about what the the determinant and cross product would correspond to.

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