What is the proper matrix product?

[SVN]

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations:
1) matrix addition;
2) multiplication by the undelying field elements;
3) matrix multiplication.

Is the last one really obvious? And is it correct? The very definition of algebra implies that any two objects belonging to it can be multiplied. This is certainly not true for matrix multiplication. I see matrix multiplication as abstraction and generalisation of the idea of the inner product (please correct me, if I am wrong). If so, the proper algebraic multiplication operation for matrices would be the Kronecker (direct) product (equivalent and generalisation of the outer product concept). It means, matrix multiplication should be considered as additional structure imposed of matrix algebra that is, strictly speaking, unnecessary for existence for the algebra of matrices, is it?

 

[fresh_42]

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations:
1) matrix addition;
2) multiplication by the undelying field elements;
3) matrix multiplication.

Is the last one really obvious? And is it correct? The very definition of algebra implies that any two objects belonging to it can be multiplied. This is certainly not true for matrix multiplication.

This is certainly true for matrix multiplication! Matrices are the coordinate form of linear transformations. The consecutive application of those transformations corresponds to matrix multiplication.

I see matrix multiplication as abstraction and generalisation of the idea of the inner product (please correct me, if I am wrong).

You are wrong. E.g. matrix multiplication can be defined on areas which do not allow inner products.

If so, the proper algebraic multiplication operation for matrices would be the Kronecker (direct) product (equivalent and generalisation of the outer product concept). It means, matrix multiplication should be considered as additional structure imposed of matrix algebra that is, strictly speaking, unnecessary for existence for the algebra of matrices, is it?

The word algebra implies that binary multiplication. Otherwise it is no algebra.

 

[SVN]

@fresh_42
I am afraid I am missing your point. We can multiply only those matrices that have equal numbers of rows and columns. For example we can multiply a matrix 2x3 by another matrix 3x4. But how should we multiply 2x3 by 4x4? It is not defined, is it? So, how can one refer to set of matrices as algebra, if some elements can not be multiplied?

Kronecker product on the other hand is defined for any two matrices.

 

[Mark44]

We can multiply only those matrices that have equal numbers of rows and columns. For example we can multiply a matrix 2x3 by another matrix 3x4. But how should we multiply 2x3 by 4x4? It is not defined, is it? So, how can one refer to set of matrices as algebra, if some elements can not be multiplied?

What is the relevant text in your book that justifies the matrix multiplication item in your list? Does it specify that the "set of matrices" are all n X n? If so, matrix multiplication is defined. If the matrices are not square, then multiplication of A and B is defined only if A is m x n, and B is n x p; i.e., the number of columns of the left matrix equals the number of rows of the right matrix. I'm sure you're aware of this.

 

[fresh_42]

You are right. We can multiply matrices if their sizes fit. If we talk about algebras, then it is the multiplication on the same space, i.e. the matrices are all square matrices of the same size.

The Kronecker product is the tensor product on a tensor algebra. The matrices are the vectors, and the algebra is the direct sum of all possible tensors: $T(V)=\bigoplus_{k=0}^\infty V^{\otimes k}$. It is an infinite dimensional, graded algebra. Strictly seen we would have infinitely many zero coordinates, which we - of course - do not write out. But the vectors are infinitely large and we consider only the non zero parts. That's why there can be differently many such non zero elements. But the vectors do not have a different size. The matrix notation in the Kronecker product is just for convenience.

 

[mathwonk]

note that addition of matrices is also not defined unless they are the same size.

 

[SVN]

@fresh_42

I see matrix multiplication as abstraction and generalisation of the idea of the inner product (please correct me, if I am wrong).

You are wrong. E.g. matrix multiplication can be defined on areas which do not allow inner products.

Let me explain myself. Of course, I did not mean the inner product operation and matrix multiplication to be the same.

Let's say we have two vectors (vector and covector to be precise) that we will regard as matrices with one row and one column. So, by definition of inner product we have
\begin{gather}
\begin{vmatrix}
x_1 & x_2 & x_3
\end{vmatrix}
\begin{vmatrix}
y_1 \\
y_2 \\
y_3
\end{vmatrix}
=
\begin{vmatrix}
x_1y_1 + x_2y_2 + x_3y_3
\end{vmatrix}
\end{gather}
Now we consider simple matrix multiplication:
\begin{gather}
\begin{vmatrix}
x_{11} & x_{12} & x_{13} \\
x_{21} & x_{22} & x_{23} \\
\end{vmatrix}
\begin{vmatrix}
y_1 \\
y_2 \\
y_3
\end{vmatrix}
=
\begin{vmatrix}
x_{11}y_1 + x_{12}y_2 + x_{13}y_3 \\
x_{21}y_1 + x_{22}y_2 + x_{23}y_3
\end{vmatrix}
\end{gather}
You can see that we have two inner products in the second case packed into 2x1 matrix. That is why I used the word «generalisation». Apart from that, vectors are geometrical objects and matrices are merely arrays of numbers, so I used the term «abstraction». It looks like matrix multiplication is related somehow with the inner product.

I am not a professional mathematician, as, probably, you are, so I do not know the right term to denote the relationship of these two concepts, but, maybe, you can shed light on it. I would very much appreciate that.

 

[fresh_42]

Sure, $v^\tau\cdot v$ is a matrix multiplication, not automatically an inner product. Hence your statement comes down to: matrix multiplication is a generalization of matrix multiplication.

 

[SVN]

@fresh_42
What is wrong with regarding $v^T\cdot v$ as an inner product automatically?

 

[fresh_42]

The fact that you said nothing about the scalar domain: $1^2+1^2=0 \in \mathbb{Z}_2.$