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Is there any way to find the eigenvalues of a matrix in terms of a variable?

asked 2020-01-10 04:36:02 -0500

Captcha gravatar image

I have the following matrix

M=

2n-1 & n-1 & n

1 & 2n-3 & 0

1& 0 & 1

Here $n$ is a variable

I want to find its eigenvalues.

Is there any way to find the eigenvalues of this matrix in terms of $n$ in sagemath.

I even cant input a matrix in terms of a variable.

Can someone please help me out?

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answered 2020-01-10 07:41:44 -0500

eric_g gravatar image

updated 2020-01-10 07:52:44 -0500

Like this:

n = var('n')
M = matrix([[2*n-1, n-1, n],
            [1, 2*n-3, 0], 
            [1, 0, 1]])
M.eigenvalues()

The outcome is a list of three eigenvalues:

[-1/6*((4*n - 3)^2 - 12*n^2 + 18*n)*(-I*sqrt(3) + 1)/((4*n - 3)^3 - 9*(2*n^2 - 3*n)*(4*n - 3) + 27*n^2 + 9*sqrt(1/3)*sqrt(-(16*n^4 - 75*n^3 + 140*n^2 - 126*n + 54)*n) - 81*n + 54)^(1/3) - 1/6*((4*n - 3)^3 - 9*(2*n^2 - 3*n)*(4*n - 3) + 27*n^2 + 9*sqrt(1/3)*sqrt(-(16*n^4 - 75*n^3 + 140*n^2 - 126*n + 54)*n) - 81*n + 54)^(1/3)*(I*sqrt(3) + 1) + 4/3*n - 1,
 -1/6*((4*n - 3)^2 - 12*n^2 + 18*n)*(I*sqrt(3) + 1)/((4*n - 3)^3 - 9*(2*n^2 - 3*n)*(4*n - 3) + 27*n^2 + 9*sqrt(1/3)*sqrt(-(16*n^4 - 75*n^3 + 140*n^2 - 126*n + 54)*n) - 81*n + 54)^(1/3) - 1/6*((4*n - 3)^3 - 9*(2*n^2 - 3*n)*(4*n - 3) + 27*n^2 + 9*sqrt(1/3)*sqrt(-(16*n^4 - 75*n^3 + 140*n^2 - 126*n + 54)*n) - 81*n + 54)^(1/3)*(-I*sqrt(3) + 1) + 4/3*n - 1,
 4/3*n + 1/3*((4*n - 3)^2 - 12*n^2 + 18*n)/((4*n - 3)^3 - 9*(2*n^2 - 3*n)*(4*n - 3) + 27*n^2 + 9*sqrt(1/3)*sqrt(-(16*n^4 - 75*n^3 + 140*n^2 - 126*n + 54)*n) - 81*n + 54)^(1/3) + 1/3*((4*n - 3)^3 - 9*(2*n^2 - 3*n)*(4*n - 3) + 27*n^2 + 9*sqrt(1/3)*sqrt(-(16*n^4 - 75*n^3 + 140*n^2 - 126*n + 54)*n) - 81*n + 54)^(1/3) - 1]

Let us check that the first element of this list is indeed an eigenvalue of M:

lamb = M.eigenvalues()[0]
det(M - lamb*identity_matrix(SR, 3)).simplify_full()

The outcome is

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Comments

From the output it looks that the first two eigenvalues are imaginary , but if you put any particular value of $n$ the eigenvalues are real and hence the result does not match

Captcha gravatar imageCaptcha ( 2020-01-12 07:14:51 -0500 )edit

what shall i do?

Captcha gravatar imageCaptcha ( 2020-01-12 07:15:53 -0500 )edit

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Asked: 2020-01-10 04:36:02 -0500

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Last updated: Jan 10