Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of naturals.

I quote the following:

A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the non negative integers and with determinant 1. To our great surprise we discovered that: there are only two primes, and any member of M (except I) can be uniquely factorized into a product of those two.

[EDIT (PLC): For some reason, there seems to be some confusion on which two matrices are in question. They are as follows:]

$$ P = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right),\ \ Q = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) $$

My question is whether there are any other resources on prime matrices and if there has been any generalization beyond 2x2 matrices. What mathematics will I need to pursue the subject further?

I have only three journal articles:

1.On prime matrices over distributive lattices

2.Algorithm for obtaining the proper relatively prime matrices of polynomial matrices

3.Prime matrices

Please specify whether I can get the article/journal or book for free.

Thanks.

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of naturals.

I quote the following:

A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the non negative integers and with determinant 1. To our great surprise we discovered that: there are only two primes, and any member of M (except I) can be uniquely factorized into a product of those two.

[EDIT (PLC): For some reason, there seems to be some confusion on which two matrices are in question. They are as follows:]

$$ P = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right),\ \ Q = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) $$

My question is whether there are any other resources on prime matrices and if there has been any generalization beyond 2x2 matrices. What mathematics will I need to pursue the subject further?

I have only three journal articles:

1. Marenich, V. E., On prime matrices over distributive lattices, J. Math. Sci., New York 164, No. 2, 260-271 (2010); translation from Fundam. Prikl. Mat. 14, No. 7, 157–173 (2008). Zbl 1288.15041.

2. Wang, Qianhua; Zhang, Zhongjun, Algorithm for obtaining the proper relatively prime matrices of polynomial matrices, Int. J. Control 46, 769–784 (1987). Zbl 0634.65030.

3. Rivett, P. F.; Mackinnon, N. I. P., Prime matrices. Math. Gaz. 70, No. 454, 257–259 (1986).

Please specify whether I can get the article/journal or book for free.

Thanks.

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of naturals.

I quote the following:

A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the non negative integers and with determinant 1. To our great surprise we discovered that: there are only two primes, and any member of M (except I) can be uniquely factorized into a product of those two.

[EDIT (PLC): For some reason, there seems to be some confusion on which two matrices are in question. They are as follows:]

$$ P = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right),\ \ Q = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) $$

My question is whether there are any other resources on prime matrices.

I have only three journal articles:

1.On prime matrices over distributive lattices

2.Algorithm for obtaining the proper relatively prime matrices of polynomial matrices

3.Prime matrices

Please specify whether I can get the article/journal or book for free.

Thanks.

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of naturals.

I quote the following:

A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the non negative integers and with determinant 1. To our great surprise we discovered that: there are only two primes, and any member of M (except I) can be uniquely factorized into a product of those two.

[EDIT (PLC): For some reason, there seems to be some confusion on which two matrices are in question. They are as follows:]

$$ P = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right),\ \ Q = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) $$

My question is whether there are any other resources on prime matrices and if there has been any generalization beyond 2x2 matrices. What mathematics will I need to pursue the subject further?

I have only three journal articles:

1.On prime matrices over distributive lattices

2.Algorithm for obtaining the proper relatively prime matrices of polynomial matrices

3.Prime matrices

Please specify whether I can get the article/journal or book for free.

Thanks.