We know the necessary and sufficient conditions for a matrix to be idempotent, that is, a square matrix A is idempotent if and only if ker(A) = Im(I - A). Does anyone know about the necessary and sufficient condition for a matrix to be tripotent?
A3 =A imply that the minimal polynomial of A is X(X-1)(X+1) which is equivalent to:
A is a diagonalizable matrix with 0, -1, 1 its eigenvalues. So we can give the equvalent conditions:
A square matrix A is tripotent if and only if it is similar to Diag(0, -1, 1).
Remark:
don't think that the matrix is of order 3 only. no, not necessary, because the eigenvalues 0, -1, 1 are counted with their multiplicities in the characteristic polynomial of A. So the question may be will be on the multiplicity of each of these eigenvalues. Me too I wonder?
I think we will have infinitely many classes of matrices related to the multiplicities of these eigenvalues.
Dear Prof
A square matrix A of order n is said to be tripotent if A^3 = A
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Dr/Tarek Ibrahim ( T. F. Ibrahim )
Associate professor
¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia
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Dear Hanifa,
The following 2 x 2 matrix
A = [-1 0
0 0]
is a tripotent matrix but it is not similar to Diag(0, -1, 1).
Yes you have right. Thank you very much for this example, so we correct:
The minimal polynomial of A divides X(X-1)(X+1). We can also see any matrix of that shape with higher order.
Thank you again. Don't hesitate to correct me.
Addition to George's note:
3. S. Krishnamoorthy and T. Rajagopalan, k-idempotency of linear combination of an idempotent matrix and a tripotent matrix that commute, Indian J. Pure Appl. Math., 42(2), 2011, pp.99–108
i hope I will help you.
Read about not only tripotent, but generally k-potent matrices.
Greetings!
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