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As a thought experiment, I was exploring the possibility of replacing scalar values in a square matrix with vectors instead so as to make a "vector matrix", and I've made a few findings that I'd like to share. I don't know if this concept already exists or if there are useful applications for it, so I wanted to receive some feedback regarding it. To begin, say for example you have a 2$\times$2 matrix with the form:

\begin{vmatrix} \vec{a} & \vec{b} \\ \vec{c} & \vec{d} \end{vmatrix}

The determinant of such a matrix can then be defined through the dot products of the vectors (assuming the vectors each have the same number of coordinates). This "dot product determinant" of the 2$\times$2 matrix would be $\vec{a}\cdot\vec{d}$ - $\vec{b}\cdot\vec{c}$, which would yield a scalar as a result. Note that switching around $\vec{b}$ and $\vec{c}$ in the matrix would not result in a different solution for the determinant since dot products are commutative. Also, if this were a 1$\times$1 matrix or a 3$\times$3 matrix, the determinant would be a vector instead of a scalar. It seems that for any n$\times$n "vector matrix" where n is even, the "dot product determinant" will be a scalar, whereas if n is odd, the "dot product determinant" will be a vector.

It is also possible to define a determinant through the cross products of the vectors, although this would probably be limited to vectors in 3D space, as far as I know. Given the assumption that all the vectors are in 3D space, the "cross product determinant" of the above 2$\times$2 matrix can be defined as $\vec{a}\times\vec{d}$ - $\vec{b}\times\vec{c}$, which would yield a vector as a result. In fact, for any n$\times$n "vector matrix" with all vectors in 3D space, the "cross product determinant" would always be a vector. Also, switching around $\vec{b}$ and $\vec{c}$ in the matrix would typically result in a different solution for the determinant since cross products are not commutative.

My question is two-fold: Can someone tell me if this concept of a "vector matrix" already exists? If it does exist, are there existing applications for it? If it doesn't, do any applications come to mind?

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Yes, there are, and they have certain useful applications in theory of quantum mechanics and deep machine learning.

Consider a matrix $\matrix{A}$ and vector $\vec{B}$.

The expression $A \otimes B$ is called the tensor, or also Kronecker product I believe.

It is a matrix with each element multiplied by the vector.

https://www.math3ma.com/blog/the-tensor-product-demystified

Here's a useful link that explains the concept you're talking about.

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Sure you can define one but it is important to then clearly specify what properties the said object would have. It certainly would not act or behave like a standard matrix —which is a representation of a linear transformation (from one vector space to another) in a specified basis.

Specifically, how would you define the product of two “vector matrices”? Say you go for a dot product on the lines of normal matrix multiplication. The end result will not be a “vector matrix” but one of scalars. I am not saying you will be prohibited from defining it that way. But if you do then it certainly won’t act or behave like a standard matrix.

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