A square matrix A such that An+1 = A is called a periodic matrix.
The minimum positive integer n with such property is called the period of the matrix A.
The following properties are direct from the definition:
(1) If A is nonsingular n - periodic matrix, then An = I.
(2) If A is nxn circulant matrix, then A is n - periodic matrix.
(3)The companion matrix of the polynomial p(x) = xn+1 - x is periodic matrix of period n.
(4) If A is nonsingular n - periodic matrix, then A⁻¹ is nonsingular
n -periodic matrix.
Are there any updated results or applications related to the periodic matrices?
Dear Issam, I just add some results, I do not think new, I just deduced them from the definition above,
The minimal polynomial is M(X)=X^{n+1}-X=X(X^{n}-1). We have these results:
1) a n-periodic matrix is diagonalizable on the complex field. From this result, we have
2) A can be represented as a direct sum of a zero matrix and a diagonal matrix with eigenvalues n-roots of unity (which make it so close to roots of unity).
3) So the applications of roots of unity may be applied to this kind of matrices.
4) if we define a primitive matrix to be the matrix (u_i), where u_i is a primitive root of unity (eigenvalue of A) , then, there are a number of \phi(n) primitive matrices in the decomposition of A
5) from 4), if we denote by A(n) the diagonal matrix constitute of the all nth roots of unity, then A is a direct sum of zero matrix and A(n). From the property of roots of unity set, we get A is a direct sum of zero matrix with the direct sums of all A(d) for d divides n.
May be the last representation of the n-periodic matrix can help in operator theory, like reduction
Dear Hanifa,
Thank you for your answer.
Concerning your Notes:
(1) A has distinct eigenvalues, hence diagonalizable.
(2) Is not clear, notice that A may be a singular or nonsingular matrix.
(3) Are there new applications?
(4)&(5) If you show a numerical example, this help to get your claim.
Best regards
The periodic matrices A are special subclass of power bounded matrices.
@ Wiwat,
I think you want to direct my attention to the results herein
https://www.researchgate.net/.../268072570_A_note_on_power_bounded_matrices
Best regards
Dear Issam, I am sorry for late, I just now I see your answer.
i) first case: If the matrix is singular, then, zero is one of its eigenvalues and ( X(X^n -1) is the minimal polynomial, and in this case of course, the order ( size) of the matrix is greater than n. Let E/kerA be the quotient vector space, then the induced operator A' of A on this space is a root of unity.
ii) second case: If the matrix A is nonsingular, then zero is not an eigenvalue, so the polynomial P(X)=X(X^n -1) is not even its characteristic polynomial, while the matrix A satisfies P(A)=0, so its characteristic polynomial should divide P(X), which yields to the order (the size) of A should be less or equal n. In this case, some of the roots of unity are the eigenvalues of A. Consequently, A is a root of unity of some integer k=size of A.
Please correct me if my analysis is wrong and thanks in advance.
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