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Re: Determinant and Characteristic Polynomial not working properly

• To: mathgroup at smc.vnet.net
• Subject: [mg86399] Re: [mg86343] Determinant and Characteristic Polynomial not working properly
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 10 Mar 2008 02:05:19 -0500 (EST)
• References: <200803091000.FAA12297@smc.vnet.net>

On 9 Mar 2008, at 11:00, Sebastian Meznaric wrote:

> Recently I reported a problem with the symbolic evaluation of
> eigenvalues of a Hermitian matrix. This time I also have a Hermitian
> matrix, for which Eigenvalues works as expected but the
> CharacteristicPolynomial and Det do not give the right results. Here
> goes
>
> The matrix is
>
> mat = {{-1, 0, 1/Sqrt[3], 0, 0, 2 Sqrt[2/3], 0, 0, 0, 0, 0, 0, 0,
>  0}, {0, -1, 0, 1/Sqrt[3], 0, 0, 2 Sqrt[2/3], 0, 0, 0, 0, 0, 0,
>  0}, {1/Sqrt[3], 0, -5/3, 4/Sqrt[3], 2 Sqrt[2/3], (2 Sqrt[2])/3, 0,
>  0, 0, 0, 0, 0, 0, 0}, {0, 1/Sqrt[3], 4/Sqrt[3], 1, -(2 Sqrt[2])/3,
>  0, (2 Sqrt[2])/3, 0, 0, 0, 0, 0, 0, 0}, {0, 0,
>  2 Sqrt[2/3], -(2 Sqrt[2])/3, 2/3, 0, 0, (5 Sqrt[2])/3, Sqrt[10/3],
>  0, 0, 0, 0, 0}, {2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0, 0, -41/6, 4/Sqrt[
>  3], 2 Sqrt[2/3], 0, Sqrt[15]/2, 0, 0, 0, 0}, {0, 2 Sqrt[2/3], 0, (
>  2 Sqrt[2])/3, 0, 4/Sqrt[3], -25/6, -(2 Sqrt[2])/3, 0, 0, Sqrt[15]/2,
>   0, 0, 0}, {0, 0, 0, 0, (5 Sqrt[2])/3,
>  2 Sqrt[2/3], -(2 Sqrt[2])/3, -1, -Sqrt[5/3]/2, 0, 0, Sqrt[15]/2, 0,
>  0}, {0, 0, 0, 0, Sqrt[10/3], 0, 0, -Sqrt[5/3]/2, 2, 0, 0, 0, Sqrt[
>  15]/2, 0}, {0, 0, 0, 0, 0, Sqrt[15]/2, 0, 0, 0, -5/2, 4/Sqrt[3],
>  2 Sqrt[2/3], 0, 0}, {0, 0, 0, 0, 0, 0, Sqrt[15]/2, 0, 0, 4/Sqrt[3],
>  1/6, -(2 Sqrt[2])/3, 0, 0}, {0, 0, 0, 0, 0, 0, 0, Sqrt[15]/2, 0,
>  2 Sqrt[2/3], -(2 Sqrt[2])/3, -2/3, Sqrt[3/5]/2, 3 Sqrt[2/5]}, {0, 0,
>   0, 0, 0, 0, 0, 0, Sqrt[15]/2, 0, 0, Sqrt[3/5]/2, 9/5, (3 Sqrt[6])/
>  5}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3 Sqrt[2/5], (3 Sqrt[6])/5, 6/
>  5}}
>
> It is Hermitian so there should be no imaginary eigenvalues. Now
> Factor[CharacteristicPolynomial[ham52, x]]
>
> 1/69984000(-3439853567779 - 24078974976000 x - 56470929408000 x^2 -
>  39558316032000 x^3 + 27303837696000 x^4 + 35330162688000 x^5 -
>  5536014336000 x^6 - 8814624768000 x^7 + 425502720000 x^8 +
>  954021888000 x^9 + 10077696000 x^10 - 47029248000 x^11 -
>  2519424000 x^12 + 839808000 x^13 + 69984000 x^14)
>
> Now we do
> N[Solve[CharacteristicPolynomial[ham52, x] == 0, x]]
> (note we are doing Solve first and then N) and we get
> {{x -> -9.81623}, {x -> -4.34022}, {x -> -2.72532}, {x -> -0.62248}, \
> {x -> -0.365633}, {x -> 2.}, {x -> 2.96246}, {x ->
>   4.90719}, {x -> -4.34015 -
>    0.0000367756 \[ImaginaryI]}, {x -> -4.34015 +
>    0.0000367756 \[ImaginaryI]}, {x -> -0.622083 -
>    0.000229308 \[ImaginaryI]}, {x -> -0.622083 +
>    0.000229308 \[ImaginaryI]}, {x ->
>   2.96235- 0.0000632158 \[ImaginaryI]}, {x ->
>   2.96235+ 0.0000632158 \[ImaginaryI]}}
>
> N[Eigenvalues[mat]]
>
> {-9.81623, 4.90719, -4.34017, -4.34017, -4.34017, 2.96239, 2.96239, \
> 2.96239, -2.72532, 2., -0.622216, -0.622216, -0.622216, -0.365633}
>
> and Eigenvalues[N[mat]]
> {-9.81623, 4.90719, -4.34017, -4.34017, -4.34017, 2.96239, 2.96239, \
> 2.96239, -2.72532, 2., -0.622216, -0.622216, -0.622216, -0.365633}
>

Yes, indeed, there must be a bug in in the way Det of symbolic  
matrices is comuted. One can get the correct charctristic polynomial  
by using significance arithmetic:

f = Factor[Rationalize[CharacteristicPolynomial[N[mat, 10], x], 0]]
  (x - 2)*(x^3 + 2*x^2 - 12*x - 8)^3*(x^4 + 8*x^3 - 32*x^2 - 144*x - 48)

  x /. NSolve[f == 0, x]
{-9.81623254755391, -4.340172973252067,
    -4.340172973252067, -4.340172973252067,
    -2.725323288781692, -0.6222156349319639,
    -0.6222156349319639, -0.6222156349319639,
    -0.3656331685835141, 2., 2.9623886081840314,
    2.9623886081840314, 2.9623886081840314,
    4.907189004919114}

Also, as you correclty note, Det gives the wrong answer:

  Det[mat]
  -(332576637342150268561733/188956800)

One way to get the right answer is to use the LUDecomposition:

{lu, p, c} = LUDecomposition[mat];
u = lu SparseArray[{i_, j_} /; j >= i -> 1, Dimensions[mat]];
  Det[u]
  -49152

which agrees with

  FullSimplify[Times @@ Eigenvalues[mat]]
  -49152

Andrzej Kozlowski

• References:
  • Determinant and Characteristic Polynomial not working properly
    • From: Sebastian Meznaric <meznaric@gmail.com>