Is there some kind of dot product and cross product of two matrices?

Question

We know the multiplication of two matrices is doing the dot product of each row vector and column vector.

Then, what about the matrix itself, which is contains row vectors and column vectors? is there an operation such dot product and cross product of these two matrices?

i mean, is it possible doing the dot product and cross product on matrices?

Or those dot and cross product just work on a vector?

Answer 1

Obviously there is no literal dot and cross product for matrices, because dot and cross product are concepts relating to vectors in a certain vector space. However, if we specify which properties of the dot and cross product are "crucial", then we can indeed find operations of matrices which satisfy these "crucial" properties too.

For the dot product, one such way to define a "crucial" property is by the concept of the inner product. Given a vector space $V$, if there is an operation taking two elements $u,v\in V$ and maps it to an element of a field (most usually the real numbers $\mathbb R$), given by $\langle u,v\rangle \in \mathbb R$, and furthermore this operation satisfies certain properties, then it is said to be an inner product. These properties are listed in the Wikipedia page: essentially, we require

1. Conjugate symmetry. $\langle u,v\rangle=\overline{\langle v,u\rangle}$ for all $x,y\in V$. (If the inner product is real valued, then this is just $\langle u,v\rangle=\langle v,u\rangle$.)
2. Linearity in the first argument. $\langle au,v\rangle=a\langle u,v\rangle$ for all $u,v\in V$ and scalar $a$ in the underlying field, and $\langle u_1+u_2,v\rangle=\langle u_1,v\rangle+\langle u_2,v\rangle$ for all $u_1,u_2,v\in V$.
3. Positive-definite. $\langle v,v\rangle>0$ for all nonzero $v\in V$.

I will not go into the deeper reasons why we require these axioms or find them useful, but in essence this is what we have distilled to be an important property of the dot product (you can verify that the dot product is indeed an inner product). Hence any inner product is in this sense a generalisation of the dot product into other contexts, which share similar properties. In the case where we are working with matrices, there is the so-called Frobenius inner product.

The cross product is harder to generalise. Even working just within vectors and not going into matrices, the cross product doesn't really make sense outside of dimensions $3$ and $7$ (the reason is explained here). However, there is some way to generalise the cross product if we throw away some familiar properties, and the most useful such generalisation is the concept of the wedge product.

In conclusion, no literal dot and cross products exist for matrices because they are concepts defined for vectors. But if we decide to keep some properties (by using them as defining axioms), and are okay with throwing away some others, then we can end up with analogues or generalisations of the dot and cross product for matrices. How useful these are and whether they can be considered "truly" faithful to the dot and cross product will depend on what you're trying to do.