Let A be a 3*3 matrix. It is very hard to calculate its determinant (it is not decidable). I know that its determinant is zero infinitely many times in a discrete manner (it is zero in an infinite discrete set of points). I want to find a dynamical system depending on A (this system can linear or nonlinear, discrete mapping or continuous time system) such that some properties of this system leading to the fact that the determinant is zero infinitely many times. I try all statements equivalent to the determinant but without any sucess. I want to see a dynamical property that forces the determinant to be zero.
Zero determinant means that zero eigenvalue of the matrix exists. Hence, it is more convenient to use the basis from eigenvectors/ It is natural and conventional. Did you use this representation of the matrix?
Yes, I have used all known statements equivalent to zero determinant.
If the system represented by A can be a closed loop system, then, you can design a control law such that detA=0. For instance if dx/dt = Fx+Bu represents a linear system and a static state feedback u=-Kx is used, hence, the objective is to design K such that det(F-BK)=0 that allows many possibilities.
Hello,
Let us assume that the matrix is NxN, where N is any integer number, e.g. N=3. If the determinant of this matrix will be equal to zero, then, at least one of the eigenvalues (of this matrix, of course) will be equal to zero. Now you can think about an ODE system with such a matrix.
Dear questioner, you are a student, aren’t you? However, I’m not your teacher, unfortunately.
Let me ask you one question for your pleasure, hopefully. Let a real (scalar) function F(x) be continuous at all points x>0 and the derivative over the coordinate x exists and is zero almost everywhere (e.g., at all irrational points). Moreover, the function is strictly monotone decreasing. How it could be? It should be interesting for you to find a few examples of using such functions in physics, dynamical systems, etc. By the way, the answer could be useful for you with your problem.
Please, don’t send to me your answer.
In addition to looking at the eigenvalues, you can look at the LU factorization of the square matrix. Assuming we compute it so that L has unit diagonals, the rank deficiency of the matrix can be measured by the number of diagonal elements of U that are zero. Perhaps Zeraoulia has already tried this, but we might as well build the list of all known equivalent statements to the zero determinant.
Dear Ed Klotz:
Yes, I have not tries d that. I can add this link for equivalent statements:http://www.applmath.engr.scu.edu/~garrison/AMTH245-F06/EQUIVALENT%20STATEMENTS.html
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