Det
Usage Message: Det[m] gives the determinant of the square matrix m. Attributes[Det] = {Protected} Options: Modulus
-> 0
Notes: For small matrices (smaller than about 11 by 11), Det[m] is computed using simple cofactor expansion. For larger matrices, cofactor expansion can be prohibitively time-consuming. Instead, for larger matrices, the Det function uses Gaussian elimination or partial row reduction to reduce the complexity of the calculation before computing the determinant. This optimization provides for much faster computation of determinants. The form of the results from Det[m] for large symbolic matrices will not always be identical to the form that would be given by cofactor expansion. The most commonly encountered example of this difference is in computation of the characteristic polynomial for a large matrix (larger than about 11 by 11). The result will sometimes be given as a rational function, reflecting the initial transformations that were used to speed up the calculation, rather than as a polynomial.
In[1]:= m = N[Table[Sin[x y], {x, 14}, {y, 14}]] ;
In[2]:= d = Det[m - x IdentityMatrix[14]]
119 -198 -198
Out[2]= (-1.21726 10 (1.32519 10 + 1.6948 10 x -
-198 2 -198 3 -197 4
> 5.7317 10 x - 7.636 10 x + 1.00354 10 x +
-197 5 -197 6 -197 7
> 1.35802 10 x - 1.02941 10 x - 1.32725 10 x +
-198 8 -198 9 -198 10
> 7.18524 10 x + 8.03595 10 x - 3.61125 10 x -
-198 11 -198 12 -199 13
> 3.16315 10 x + 1.30586 10 x + 8.21605 10 x -
-199 14 -199 15 -200 16
> 3.30445 10 x - 1.40429 10 x + 5.66687 10 x +
-200 17 -201 18 -201 19
> 1.55058 10 x - 6.362 10 x - 1.05921 10 x +
-202 20 -203 21 -203 22
> 4.44238 10 x + 4.05856 10 x - 1.74284 10 x -
-205 23 -205 24
> 6.6597 10 x + 2.928 10 x )) /
-17 -18 -18 2 5
> (1.54539 10 + 5.71835 10 x - 8.13562 10 x )
From this result you can in principle get the characteristic polynomial (and a remainder polynomial with numerically negligible coefficients) using
PolynomialDivision
. For reasons of numerical stability, this method is often appropriate only for high-precision matrices.
In[3]:= {num, den} = {Numerator[d], Denominator[d]} ;
In[4]:= {cp, rem} = PolynomialDivision[num, den, x]
2 3 4
Out[4]= {-183008. + 104539. x + 366989. x - 112690. x - 230950. x +
5 6 7 8 9
> 46597.1 x + 69568. x - 9566.38 x - 11424.3 x + 1037.15 x +
10 11 12 13 14
> 1048.91 x - 56.8254 x - 50.6084 x + 1.2399 x + 1. x ,
-92 -92 -92 2 -92 3
> -1.61499 10 - 3.09815 10 x + 2.0075 10 x + 6.05247 10 x -
-93 4 -92 5 -93 6
> 6.39704 10 x - 4.80003 10 x - 1.07117 10 x +
-92 7 -94 8 -93 9
> 1.87267 10 x + 7.11615 10 x - 3.03747 10 x }
In examples such as this, where the symbolic part of the matrix occurs in a particularly simple form, there are often better ways to do the calculation rather than direct computation of a symbolic determinant. For example, you can construct the characteristic polynomial using the eigenvalues of the matrix (which are the same as the roots of the characteristic polynomial computed above).
In[5]:= ev = Eigenvalues[m] //Sort
Out[5]= {-3.34798, -3.29011, -3.26555, -3.2125, -2.35978, -2.06242, -1.09661,
> 0.739625, 1.97485, 2.28356, 2.4223, 3.22977, 3.32804, 3.41689}
In[6]:= NRoots[cp == 0, x]
Out[6]= x == -3.34798 || x == -3.29011 || x == -3.26555 || x == -3.2125 ||
> x == -2.35978 || x == -2.06242 || x == -1.09661 || x == 0.739625 ||
> x == 1.97485 || x == 2.28356 || x == 2.4223 || x == 3.22977 ||
> x == 3.32804 || x == 3.41689
Another approach is to compute values for the characteristic polynomial using interpolation. Here is an example of the use of interpolation to compute the eigenvalue near x == 0.739625. This approach has the advantage of never requiring a potentially time-consuming evaluation of a symbolic determinant. The example here, which involves the computation of 51 numerical determinants, will normally be faster than a single evaluation of the corresponding symbolic determinant.
In[7]:= cpi[x_] = Interpolation[
Table[{t, Det[m - t IdentityMatrix[14]]}, {t, 0, 1, .02}]][x]
Out[7]= InterpolatingFunction[{{0, 1.}}, <>][x]
In[8]:= FindRoot[cpi[x], {x, 0, 1}]
Out[8]= {x -> 0.739625}
Known Bugs The Det function in Version 3.0 of Mathematica can return a result with an incorrect sign for certain large exact matrices. This error is currently being investigated. Although a complete characterization of this error is not yet available, all of the known examples are exact matrices with dimension 12 or larger and with a matrix element -1 as the only non-zero element in some column or row. Here is an example of a result with an incorrect sign.
In[1]:= Det[DiagonalMatrix[{-1,2,3,4,5,6,7,8,9,a,b,c}]]
Out[1]= 362880 a b c
You can get a correct result in this example by computing the product of the eigenevalues.
In[2]:= Times @@ Eigenvalues[DiagonalMatrix[{-1,2,3,4,5,6,7,8,9,a,b,c}]]
Out[2]= -362880 a b c
Here is a more complicated example, which also gives an incorrect sign. The correct result can be computed using the product of the eigenvalues.
In[3]:= m = {{-1,0,0,0,0,0,0,0,0,0,0,0},
{0,2,0,1,0,0,0,1,0,0,0,0},
{0,0,3,0,0,0,0,0,0,0,0,0},
{0,0,0,4,0,0,0,0,1,0,0,0},
{0,1,0,0,5,0,0,0,1,0,0,0},
{0,0,0,0,0,6,0,0,0,0,1,1},
{0,0,1,0,0,0,7,0,0,1,0,0},
{0,0,0,0,0,0,0,8,0,0,0,0},
{0,0,0,1,0,0,0,0,9,0,0,0},
{0,0,0,1,0,0,1,0,0,x,0,0},
{0,1,0,0,0,0,1,0,0,0,y,0},
{0,0,0,0,0,0,0,0,0,0,0,z}};
In[4]:= Det[m]
Out[4]= 50400 (-y + 7 x y) z
In[5]:= Times @@ Eigenvalues[m] //Expand
Out[5]= 50400 y z - 352800 x y z
When additional information about this error becomes available, it will be included here, or can be obtained by contacting Technical Support
. We apologize for any difficulties caused by this error.
Additional Online Documentation:
Mathematica 3.0
http://documents.wolfram.com/v3/RefGuide/Det.html
Mathematica 4.0
http://documents.wolfram.com/v4/RefGuide/Det.html
Questions or comments? Send email to support@wolfram.com.
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