ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 24 Feb 2021 15:17:19 +0100How one can find a matrix with given condition?https://ask.sagemath.org/question/55877/how-one-can-find-a-matrix-with-given-condition/I found the following recreational mathematics problem too hard for me. Can anyone give me hints how to find a solution?
Consider a $4\times 4$ matrix with integer coefficients. Let its elements be $a_{i,j}$ for $1\leq i,j\leq 4$. Now form four sets $A_1$, $A_2$, $A_3$, $A_4$. Say that $A_1$ is the set of elements $a_{i,j}$. Also, $A_2$ is the set of elements $a_{i,j}+a_{i+1,j}+a_{i,j+1}+a_{i+1,j+1}$ i.e. the set of sums of elements of $2\times 2$-subsquares of the matrix. Similarly, denote by $A_3$ the set of sums of $3\times 3$-submatrices and $A_4$ the set containing the sum of all elements of the given $4\times 4$-matrix.
Now, I heard a rumor that one can give an example of 16 integers $a_{i,j}$ such that all integers from 1 to 25 belong to $A_1 \cup A_2 \cup A_3 \cup A_4$. How can one find such an example? It is easy to see that at least some of the elements $a_{i,j}$ must satisfy $1\leq a_{i,j}\leq 25$ but there are still so many elements that finding such a matrix by brute force seems impossible. I was wondering if genetic algorithms or simulated annealing work for such a problem but I don't have enough experience to implement that.
The best I know is that getting all integers from 1 to 24 is possible, for instance using the matrix:
-42 22 23 7
13 11 -32 14
-23 16 15 8
19 9 -22 1Jaakko SeppäläWed, 24 Feb 2021 15:17:19 +0100https://ask.sagemath.org/question/55877/generation of certain matriceshttps://ask.sagemath.org/question/37860/generation-of-certain-matrices/I'd like to create a list of roughly 100-1000 $2 \times 2$ matrices $[A_1,A_2,...,A_N]$ that have the following properties:
1. $\det A_j = 1$
2. The entries of each $A_j$ are in an imaginary quadratic integer ring, such as $\mathbb{Z}[i]$, or $\mathbb{Z}[\sqrt{-2}]$
For example, the matrix
$$\begin{bmatrix}
1&2i \\\\
0&1
\end{bmatrix}$$
fits the above specifications when the ring is $\mathbb{Z}[i]$.
I know that I probably want to run some kind of loop over the entries of the matrix, but I'm not sure how to do this. Perhaps I want to initially treat the matrices as lists of length 4, and then run an iterative loop over the lists. Then, when the above specifications are met, that list is stored somewhere else. I think I'd also like to put a bound on the "size" of the matrix entries, but that should be easy to do afterwards.
Thanks! Daniel LThu, 08 Jun 2017 19:26:31 +0200https://ask.sagemath.org/question/37860/