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Eigenvalue and eigenvectors of a 10x10 matrix

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Redeemed

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Sep 13, 2012, 3:41:56 AM9/13/12
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I want to do the eigen analysis of the matrix below
mat := {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {-(1 + K + K1), -0.1, K, 0, 0,
0, 0, 0, K1, 0},
{0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {K, 0, -(1 + 2 K + K1), -0.1, K, 0,
0, 0, K1, 0},
{0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, K, 0, -(1 + 2 K + K1), -0.1,
K, 0, K1, 0},
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, K,
0, -(1 + K + K1), -0.1, K1, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {K1, 0, K1, 0, K1, 0, K1,
0, -(1 + 4 K1), -0.1}};

I kept getting a long solution with some Root [] and #1
I do not know what I am doing wrong
Any help,
Its very urgent

Thanks

Bob Hanlon

unread,
Sep 14, 2012, 12:14:46 AM9/14/12
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mat = {
{0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
{-(1 + K + K1), -0.1, K, 0, 0, 0, 0, 0, K1, 0},
{0, 0, 0, 1, 0, 0, 0, 0, 0, 0},
{K, 0, -(1 + 2 K + K1), -0.1, K, 0, 0, 0, K1, 0},
{0, 0, 0, 0, 0, 1, 0, 0, 0, 0},
{0, 0, K, 0, -(1 + 2 K + K1), -0.1, K, 0, K1, 0},
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
{0, 0, 0, 0, K, 0, -(1 + K + K1), -0.1, K1, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
{K1, 0, K1, 0, K1, 0, K1, 0, -(1 + 4 K1), -0.1}};

Rationalize the matrix first and you will get radicals rather than Root objects.

ev1 = Eigenvalues[Rationalize[mat]] // Simplify

{(1/20)*I*(I + Sqrt[399]), (-(1/20))*I*(-I + Sqrt[399]),
(1/20)*(-1 - Sqrt[-399 - 2000*K1]),
(1/20)*(-1 + Sqrt[-399 - 2000*K1]),
(1/20)*(-1 - Sqrt[-399 - 800*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 800*K - 400*K1]),
(1/20)*(-1 - Sqrt[-399 - 400*(2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 400*(2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 - Sqrt[-399 + 400*(-2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 + 400*(-2 + Sqrt[2])*K - 400*K1])}

Alternatively,

ev2 = Eigenvalues[mat] // Rationalize // ToRadicals // Simplify

{(1/20)*(-1 - Sqrt[-399 + 400*(-2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 + 400*(-2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 - Sqrt[-399 - 400*(2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 400*(2 + Sqrt[2])*K - 400*K1]),
(1/20)*(-1 - Sqrt[-399 - 2000*K1]),
(1/20)*(-1 + Sqrt[-399 - 2000*K1]),
(1/20)*(-1 - Sqrt[-399 - 800*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 800*K - 400*K1]),
(-(1/20))*I*(-I + Sqrt[399]), (1/20)*I*(I + Sqrt[399])}

Sort[ev1] === Sort[ev2]

True

To learn about Root objects see
http://reference.wolfram.com/mathematica/ref/Root.html



Bob Hanlon

Frank K

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Sep 14, 2012, 12:15:06 AM9/14/12
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You're probably not doing anything wrong. You're asking for a symbolic solution for a 10th order polynomial. It's bound to complicated.

Ray Koopman

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Sep 14, 2012, 12:15:53 AM9/14/12
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Change .1 to 1/10

mat = {
{0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
{-(1+K+K1), -1/10, K, 0, 0, 0, 0, 0, K1, 0},
{0, 0, 0, 1, 0, 0, 0, 0, 0, 0},
{K, 0, -(1+2K+K1), -1/10, K, 0, 0, 0, K1, 0},
{0, 0, 0, 0, 0, 1, 0, 0, 0, 0},
{0, 0, K, 0, -(1+2K+K1), -1/10, K, 0, K1, 0},
{0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
{0, 0, 0, 0, K, 0, -(1+K+K1), -1/10, K1, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
{K1, 0, K1, 0, K1, 0, K1, 0, -(1+4K1), -1/10}};

Eigenvalues[mat]

{(1/20)*(-1 + I*Sqrt[399]),
(1/20)*(-1 - I*Sqrt[399]),
(1/20)*(-1 - Sqrt[-399 - 2000*K1]),
(1/20)*(-1 + Sqrt[-399 - 2000*K1]),
(1/20)*(-1 - Sqrt[-399 - 800*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 800*K - 400*K1]),
(1/20)*(-1 - Sqrt[-399 - 800*K - 400*Sqrt[2]*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 800*K - 400*Sqrt[2]*K - 400*K1]),
(1/20)*(-1 - Sqrt[-399 - 800*K + 400*Sqrt[2]*K - 400*K1]),
(1/20)*(-1 + Sqrt[-399 - 800*K + 400*Sqrt[2]*K - 400*K1])}

Eigensystem[mat] returns similar output, but too much to give here.

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