The Strange Case of the Determinant of a Matrix of 1s and -1s

By Nick Higham and Alan Edelman (MIT)

In a 2005 talk the second author noted that the MATLAB det function returns an odd integer for a certain 27-by-27 matrix composed of 1s and -1s:

>> A = edelman; % Set up the matrix.
>> format long g, format compact, det(A)
ans =
           839466457497601

However, the determinant is, from its definition, a sum of an even number (27 factorial) of odd numbers, so is even. Indeed the correct determinant is 839466457497600.

At first sight, this example is rather troubling, since while MATLAB returns an integer, as expected, it is out by 1. The determinant is computed as the product of the diagonal entries of the U factor in the LU factorization with partial pivoting of A, and these entries are not all integers. Standard rounding error analysis shows that the relative error from forming that product is bounded by nu/(1-nu), with n=27, where u \approx 1.1 \times 10^{-16} is the unit roundoff, and this is comfortably larger than the actual relative error (which also includes the errors in computing U) of 6 \times 10^{-16}. Therefore the computed determinant is well within the bounds of roundoff, and if the exact result had not been an integer the incorrect last decimal digit would hardly merit discussion.

However, this matrix has more up its sleeve. Let us compute the determinant using a different implementation of Gaussian elimination with partial pivoting, namely the function gep from the Matrix Computation Toolbox:

>> [Lp,Up,Pp] = gep(A,'p'); det(Pp)*det(Up)
ans =
           839466457497600

Now we get the correct answer! To see what is happening, we can directly form the products of the diagonal elements of the U factors:

>> [L,U,P] = lu(A); 
>> d = diag(U); dp = diag(Up); 
>> rel_diff_U_diags = norm((dp - d)./d,inf)
rel_diff_U_diags =
      7.37206353875273e-16
>> [prod(d), prod(dp)]
ans =
          -839466457497601          -839466457497600
>> [prod(d(end:-1:1)), prod(dp(end:-1:1))]
ans =
          -839466457497600          -839466457497600

We see that even though the diagonals of the two U factors differ by a small multiple of the unit roundoff, the computed products differ in the last decimal digit. If the product of the diagonal elements of U is accumulated in the reverse order then the exact answer is obtained in both cases. Once again, while this behaviour might seem surprising, it is within the error bounds of a rounding error analysis.

The moral of this example is that we should not be misled by the integer nature of a result; in floating-point arithmetic it is relative error that should be judged.

Finally, we note that numerical evaluation of the determinant offers other types of interesting behaviour. Consider the Frank matrix: a matrix of integers that has determinant 1. What goes wrong here in the step from dimension 24 to 25?

>> A = gallery('frank',24); det(A)
ans =
         0.999999999999996
>> A = gallery('frank',25); det(A)
ans =
          143507521.082525

The Edelman matrix is available in the MATLAB function available in this gist, which is embedded below. A Julia notebook exploring the Edelman matrix is available here.

function A = edelman
%EDELMAN Alan Edelman's matrix for which det is computed as the wrong integer.
% A = EDELMAN is a 27-by-27 matrix of 1s and -1s for which the
% MATLAB det function returns an odd integer, though the exact
% determinant is an even integer.
A = [%
1 1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 -1
1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 1 1 1
-1 1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1
-1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 -1
-1 1 1 -1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 -1
1 1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 -1
-1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 -1 1 -1
1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 -1
-1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 1 1 1 -1 1 1 -1 1 1 1 -1 -1
-1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1
-1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 1
1 1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1
-1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1
1 1 -1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1
-1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1
-1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1
1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1
1 1 -1 1 1 -1 1 1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1
1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1
-1 -1 -1 1 1 -1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1
1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1
-1 -1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1
-1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 1
-1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1
1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 1
1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1
1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 1 1 -1 1];
view raw edelman.m hosted with ❤ by GitHub

2 thoughts on “The Strange Case of the Determinant of a Matrix of 1s and -1s”

  1. QR decomposition both in Julia (qr or qrfact) and in Matlab ([Q,R,E] = qr(A)) seems to give a result that is closer to the exact value than using LU or SVD.

    Reply

  2. Using the quadruple precision:

    >> A = mp(gallery('frank',24));
>> det(A)
ans = 
    0.9999999999976255956641569567784582
>> log10(cond(A))
ans = 
    24.9757014216020782990166105113167

>> A = mp(gallery('frank',25));
>> det(A)
ans = 
    0.999999999996643247248220196566582
>> log10(cond(A))
ans = 
    26.38848037203079081910169501698983

    I guess in case of Frank-25 ill-conditioning becomes too high for double-precision LU. I think the most interesting question is why it still works for Frank-24?

    Out of curiosity, quadruple applied for the Edelman matrix:

    >> A = mp(edelman);
>> det(A)
ans = 
    839466457497600.0000000000000000004

>> [L,U,P] = lu(A);
>> prod(diag(U))
ans = 
    -839466457497600.0000000000000000004
    Reply

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