Determinants of binary matrices












14












$begingroup$


I was surprised to find that most $3times 3$ matrices with entries in ${0,1}$ have determinant $0$ or $pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ and three with $-2$) and these are
$$
begin{bmatrix}1 & 1 & 0\ 0 & 1 & 1\ 1 & 0 & 1end{bmatrix}
$$
and the different permutation of this.



Hadamard's maximum determinant problem for ${0,1}$ asks about the largest possible determinant for matrices with entries in ${0,1}$ and it is known that the sequence of the maximal determinant for $ntimes n$ matrices for $n=1,2,dots$ starts with 1, 1, 2, 3, 5, 9, 32, 56, 144, 320,1458 (https://oeis.org/A003432). It is even known that the number of different matrices realizing the maximum (not by absolute value) is 1, 3, 3, 60, 3600, 529200, 75600, 195955200, (https://oeis.org/A051752).




My question is: what is the distribution of determinants of all $ntimes n$ ${0,1}$-matrices?




Here is the data for (very) small $n$:
$$
begin{array}{lccccccc}
n & -3 & -2 & -1 & 0 & 1 & 2 & 3\hline
1 & & & & 1 & & & \
2 & & & 3 &10 & 3 & & \
3 & & 3 & 84 & 338 & 84 & 3 & \
4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60
end{array}
$$






linear-algebra determinants







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$endgroup$








  • 3




    $begingroup$
    Will Orrick probably has the latest data. Miodrag Zivkovic did an analysis in 2005 for n up to 9. While lower determinant values are heavily weighted, it is not known how quickly the counts for off as the determinant value grows. You can find related questions here on MathOverflow. Gerhard "Search For Determinant Spectrum Problem" Paseman, 2019.01.15.
    $endgroup$
    – Gerhard Paseman
    6 hours ago










  • $begingroup$
    Related MO questions: mathoverflow.net/questions/18636/… mathoverflow.net/questions/18547/… mathoverflow.net/questions/39786/…
    $endgroup$
    – Timothy Chow
    13 mins ago
















14












$begingroup$


I was surprised to find that most $3times 3$ matrices with entries in ${0,1}$ have determinant $0$ or $pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ and three with $-2$) and these are
$$
begin{bmatrix}1 & 1 & 0\ 0 & 1 & 1\ 1 & 0 & 1end{bmatrix}
$$
and the different permutation of this.



Hadamard's maximum determinant problem for ${0,1}$ asks about the largest possible determinant for matrices with entries in ${0,1}$ and it is known that the sequence of the maximal determinant for $ntimes n$ matrices for $n=1,2,dots$ starts with 1, 1, 2, 3, 5, 9, 32, 56, 144, 320,1458 (https://oeis.org/A003432). It is even known that the number of different matrices realizing the maximum (not by absolute value) is 1, 3, 3, 60, 3600, 529200, 75600, 195955200, (https://oeis.org/A051752).




My question is: what is the distribution of determinants of all $ntimes n$ ${0,1}$-matrices?




Here is the data for (very) small $n$:
$$
begin{array}{lccccccc}
n & -3 & -2 & -1 & 0 & 1 & 2 & 3\hline
1 & & & & 1 & & & \
2 & & & 3 &10 & 3 & & \
3 & & 3 & 84 & 338 & 84 & 3 & \
4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60
end{array}
$$






linear-algebra determinants







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    Will Orrick probably has the latest data. Miodrag Zivkovic did an analysis in 2005 for n up to 9. While lower determinant values are heavily weighted, it is not known how quickly the counts for off as the determinant value grows. You can find related questions here on MathOverflow. Gerhard "Search For Determinant Spectrum Problem" Paseman, 2019.01.15.
    $endgroup$
    – Gerhard Paseman
    6 hours ago










  • $begingroup$
    Related MO questions: mathoverflow.net/questions/18636/… mathoverflow.net/questions/18547/… mathoverflow.net/questions/39786/…
    $endgroup$
    – Timothy Chow
    13 mins ago














14












14








14


2



$begingroup$


I was surprised to find that most $3times 3$ matrices with entries in ${0,1}$ have determinant $0$ or $pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ and three with $-2$) and these are
$$
begin{bmatrix}1 & 1 & 0\ 0 & 1 & 1\ 1 & 0 & 1end{bmatrix}
$$
and the different permutation of this.



Hadamard's maximum determinant problem for ${0,1}$ asks about the largest possible determinant for matrices with entries in ${0,1}$ and it is known that the sequence of the maximal determinant for $ntimes n$ matrices for $n=1,2,dots$ starts with 1, 1, 2, 3, 5, 9, 32, 56, 144, 320,1458 (https://oeis.org/A003432). It is even known that the number of different matrices realizing the maximum (not by absolute value) is 1, 3, 3, 60, 3600, 529200, 75600, 195955200, (https://oeis.org/A051752).




My question is: what is the distribution of determinants of all $ntimes n$ ${0,1}$-matrices?




Here is the data for (very) small $n$:
$$
begin{array}{lccccccc}
n & -3 & -2 & -1 & 0 & 1 & 2 & 3\hline
1 & & & & 1 & & & \
2 & & & 3 &10 & 3 & & \
3 & & 3 & 84 & 338 & 84 & 3 & \
4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60
end{array}
$$






linear-algebra determinants







share|cite|improve this question











$endgroup$




I was surprised to find that most $3times 3$ matrices with entries in ${0,1}$ have determinant $0$ or $pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ and three with $-2$) and these are
$$
begin{bmatrix}1 & 1 & 0\ 0 & 1 & 1\ 1 & 0 & 1end{bmatrix}
$$
and the different permutation of this.



Hadamard's maximum determinant problem for ${0,1}$ asks about the largest possible determinant for matrices with entries in ${0,1}$ and it is known that the sequence of the maximal determinant for $ntimes n$ matrices for $n=1,2,dots$ starts with 1, 1, 2, 3, 5, 9, 32, 56, 144, 320,1458 (https://oeis.org/A003432). It is even known that the number of different matrices realizing the maximum (not by absolute value) is 1, 3, 3, 60, 3600, 529200, 75600, 195955200, (https://oeis.org/A051752).




My question is: what is the distribution of determinants of all $ntimes n$ ${0,1}$-matrices?




Here is the data for (very) small $n$:
$$
begin{array}{lccccccc}
n & -3 & -2 & -1 & 0 & 1 & 2 & 3\hline
1 & & & & 1 & & & \
2 & & & 3 &10 & 3 & & \
3 & & 3 & 84 & 338 & 84 & 3 & \
4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60
end{array}
$$






linear-algebra determinants




linear-algebra determinants






share|cite|improve this question















share|cite|improve this question













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edited 6 hours ago







Dirk

















asked 7 hours ago









DirkDirk

7,39143267




7,39143267








  • 3




    $begingroup$
    Will Orrick probably has the latest data. Miodrag Zivkovic did an analysis in 2005 for n up to 9. While lower determinant values are heavily weighted, it is not known how quickly the counts for off as the determinant value grows. You can find related questions here on MathOverflow. Gerhard "Search For Determinant Spectrum Problem" Paseman, 2019.01.15.
    $endgroup$
    – Gerhard Paseman
    6 hours ago










  • $begingroup$
    Related MO questions: mathoverflow.net/questions/18636/… mathoverflow.net/questions/18547/… mathoverflow.net/questions/39786/…
    $endgroup$
    – Timothy Chow
    13 mins ago














  • 3




    $begingroup$
    Will Orrick probably has the latest data. Miodrag Zivkovic did an analysis in 2005 for n up to 9. While lower determinant values are heavily weighted, it is not known how quickly the counts for off as the determinant value grows. You can find related questions here on MathOverflow. Gerhard "Search For Determinant Spectrum Problem" Paseman, 2019.01.15.
    $endgroup$
    – Gerhard Paseman
    6 hours ago










  • $begingroup$
    Related MO questions: mathoverflow.net/questions/18636/… mathoverflow.net/questions/18547/… mathoverflow.net/questions/39786/…
    $endgroup$
    – Timothy Chow
    13 mins ago








3




3




$begingroup$
Will Orrick probably has the latest data. Miodrag Zivkovic did an analysis in 2005 for n up to 9. While lower determinant values are heavily weighted, it is not known how quickly the counts for off as the determinant value grows. You can find related questions here on MathOverflow. Gerhard "Search For Determinant Spectrum Problem" Paseman, 2019.01.15.
$endgroup$
– Gerhard Paseman
6 hours ago




$begingroup$
Will Orrick probably has the latest data. Miodrag Zivkovic did an analysis in 2005 for n up to 9. While lower determinant values are heavily weighted, it is not known how quickly the counts for off as the determinant value grows. You can find related questions here on MathOverflow. Gerhard "Search For Determinant Spectrum Problem" Paseman, 2019.01.15.
$endgroup$
– Gerhard Paseman
6 hours ago












$begingroup$
Related MO questions: mathoverflow.net/questions/18636/… mathoverflow.net/questions/18547/… mathoverflow.net/questions/39786/…
$endgroup$
– Timothy Chow
13 mins ago




$begingroup$
Related MO questions: mathoverflow.net/questions/18636/… mathoverflow.net/questions/18547/… mathoverflow.net/questions/39786/…
$endgroup$
– Timothy Chow
13 mins ago










1 Answer
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This question is too hard, because easier questions are already known to be hard.



The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longstanding open problem that was just recently solved by Tikhomirov.



As you mentioned yourself, the Hadamard maximal determinant problem is unsolved, with order 15 (for the {±1} version of the problem) being difficult already. The determinant spectrum problem, mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.






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    1 Answer
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    $begingroup$

    This question is too hard, because easier questions are already known to be hard.



    The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longstanding open problem that was just recently solved by Tikhomirov.



    As you mentioned yourself, the Hadamard maximal determinant problem is unsolved, with order 15 (for the {±1} version of the problem) being difficult already. The determinant spectrum problem, mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      This question is too hard, because easier questions are already known to be hard.



      The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longstanding open problem that was just recently solved by Tikhomirov.



      As you mentioned yourself, the Hadamard maximal determinant problem is unsolved, with order 15 (for the {±1} version of the problem) being difficult already. The determinant spectrum problem, mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        This question is too hard, because easier questions are already known to be hard.



        The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longstanding open problem that was just recently solved by Tikhomirov.



        As you mentioned yourself, the Hadamard maximal determinant problem is unsolved, with order 15 (for the {±1} version of the problem) being difficult already. The determinant spectrum problem, mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.






        share|cite|improve this answer









        $endgroup$



        This question is too hard, because easier questions are already known to be hard.



        The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longstanding open problem that was just recently solved by Tikhomirov.



        As you mentioned yourself, the Hadamard maximal determinant problem is unsolved, with order 15 (for the {±1} version of the problem) being difficult already. The determinant spectrum problem, mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 4 hours ago









        Timothy ChowTimothy Chow

        34.5k11183309




        34.5k11183309






























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