What is the proper matrix product?

In summary: The first matrix has the inner product\begin{align*}& a\cdot b &= \left(a\right)^Tb\\& c\cdot d &= \left(c\right)^Td\\& e\cdot f &= \left(e\right)^TF
  • #1
SVN
49
1
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?
 
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  • #2
SVN said:
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.
 
  • #3
@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.
 
  • #4
SVN said:
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.
 
  • #5
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.
 
  • #6
note that addition of matrices is also not defined unless they are the same size.
 
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  • #7
@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.
 
  • #8
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.
 
  • #9
@fresh_42
What is wrong with regarding ##v^T\cdot v## as an inner product automatically?
 
  • #10
The fact that you said nothing about the scalar domain: ##1^2+1^2=0 \in \mathbb{Z}_2.##
 

Related to What is the proper matrix product?

1. What is a matrix product?

A matrix product is a mathematical operation that involves multiplying two matrices together. It is a fundamental operation in linear algebra and is used in various fields such as physics, engineering, and computer science.

2. How is a matrix product calculated?

To calculate a matrix product, you need to follow a specific rule known as the "row by column rule". This means that the number of columns in the first matrix must be equal to the number of rows in the second matrix. Then, you multiply the corresponding elements in each row of the first matrix with the corresponding elements in each column of the second matrix and add the results to get the value of each element in the resulting matrix.

3. What is the significance of matrix products?

Matrix products are significant because they allow us to perform operations on multiple variables simultaneously. This makes it easier to solve complex systems of equations and perform other calculations that involve multiple variables. Matrix products also have applications in data analysis, image processing, and machine learning.

4. Are there any special rules for matrix products?

Yes, there are a few special rules to keep in mind when performing matrix products. The first is the commutative property, which states that the order of multiplication does not matter for scalar multiplication. However, the order does matter for matrix multiplication. Another important rule is the associative property, which states that the grouping of matrices in a product does not change the result. Finally, the distributive property states that you can distribute a scalar multiplication across a matrix product.

5. Can any two matrices be multiplied together?

No, not all matrices can be multiplied together. As mentioned earlier, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This means that the dimensions of the matrices must be compatible for multiplication. For example, a 2x3 matrix can be multiplied by a 3x4 matrix, resulting in a 2x4 matrix. But a 2x3 matrix cannot be multiplied by a 2x4 matrix.

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