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How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$) which is similar to its inverse?

asked 2020-03-24 15:43:18 -0500

anonymous user

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updated 2020-03-25 02:13:55 -0500

How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$)which is similar to its inverse? I am trying to find one such example in Sage. Please give some idea to proceed.

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The identity matrix.

rburing gravatar imagerburing ( 2020-03-24 15:56:06 -0500 )edit

Maybe you can compare the characteristic polynomial of a matrix with the one of its inverse and try to find when the two are equal?

Sébastien gravatar imageSébastien ( 2020-03-25 02:17:05 -0500 )edit

ok.can we have at least one such example?

Kuldeep gravatar imageKuldeep ( 2020-03-25 02:47:23 -0500 )edit

On the diagonal put pairs of numbers which are each other's inverse, like 2 and 1/2.

rburing gravatar imagerburing ( 2020-03-25 03:13:09 -0500 )edit
3

Does this question have anything to do with SageMath?

vdelecroix gravatar imagevdelecroix ( 2020-03-25 07:45:03 -0500 )edit

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answered 2020-03-25 16:21:53 -0500

vdelecroix gravatar image

updated 2020-03-25 16:47:52 -0500

Let me describe three constructions of matrices which are similar to their inverses over Z. It has little to do with SageMath. That is rather linear algebra.

Construction 1: There are infinitely 2x2 matrices with integral coefficients that are similar over Z to their inverses

sage: A = matrix(2, [1,1,0,1])
sage: B = matrix(2, [1,0,1,1])
sage: P = matrix(2, [-1,0,0,1])
sage: C1 = A
sage: C2 = B
sage: C3 = A * B * A
sage: all(P * C * P**-1 == C**-1 for C in [C1, C2, C3])
True

You can just build your 6x6 matrix M as a block matrix

sage: M = matrix(ZZ, 6)
sage: M[0:2,0:2] = C1
sage: M[2:4,2:4] = C2
sage: M[4:6,4:6] = C3
sage: Q = matrix(ZZ, 6)
sage: Q[0:2,0:2] = Q[2:4,2:4] = Q[4:6,4:6] = P
sage: Q * M * Q**-1 == M**-1
True

Construction 2: The companion matrix of a symmetric polynomial

sage: x = polygen(ZZ, 'x')
sage: a = 13
sage: b = 25
sage: c = 18
sage: M = companion_matrix(X**6 + a*X**5 + b*X**4 + c*X**3 + b*X**2 + a*X + 1)
sage: S = SymmetricGroup(6)
sage: P = S('(1,6)(2,5)(3,4)').matrix()
sage: P * M * P**-1 == M**-1
True

Construction 3: Permutation matrices

sage: S = SymmetricGroup(6)
sage: M = S('(1,2,3,4)(5,6)').matrix()
sage: P = S('(1,4)(2,3)').matrix()
sage: P * M * P**-1 == M**-1
True
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1

And once you found some matrix M satisfying P * M * P**-1 == M**-1 you may create lot more by doing:

sage: Q = random_matrix(ZZ,6,6)
sage: Mnew = Q * M * ~Q
sage: Pnew = Q * P * ~Q
sage: Pnew * Mnew * ~Pnew == ~Mnew
True
Sébastien gravatar imageSébastien ( 2020-03-26 02:32:13 -0500 )edit

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Asked: 2020-03-24 15:43:18 -0500

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Last updated: Mar 25