We find eigen values in various cases, like in electrical and electronic circuits, in control systems, in data analysis etc.
Why we find it? What is the significance of eigen value? What will happen if the eigen value is complex or imaginary number?
A matrix is a table designating relations quantitatively. For example, a network.
Eigenvalues form the backbone of a matrix. Because the matrix is similar to a diagonal matrix with solely the eigenvalues, i.e. P-1AP = Λ.
(This is almost always feasible, see N. Cai, J. Cao, M.J. Khan, "Almost decouplability of any directed weighted network topology", Physica A, 2015, 436, pp. 637-645. )
The solution of the basic dynamic equation x(k+1)=Ax(k) is x(k) = Akx(0) and also x(k) = PΛkP-1x(0).
Eigenvalues dominate.
If an eigenvalue is complex, then it must have a conjugate pal.
Eigenvalue is an answer of equation or system. As an example, for schrodinger equation in a quantum well, eigenvalue means an allowed energy level for electron or hole.
See some of the suggestions in the following link:
https://www.google.com/search?q=eigenvalues+and+eigenvectors&ie=utf-8&oe=utf-8&client=firefox-b-1
In short, linear functions are common and powerful. Knowing more about them is a good thing. Best, David Booth
Eigen values are the characteristic energy values of certain wave equation
Eigenvalues of a system allow you, beside other things, assess stability of your system. If all real parts are smaller 0 your system is stable. Conjugate complex eigenvalues ell you that there the corresponding eigen-mode is a oscillation, which might be damped (real parts smaller 0), a pure undamped oscialltion (real part equal 0) or unstable (again, real part bigger than 0).
Dear Muhibul,
eigenvectors shows the characteristics of a system, e.g. imagine that E = [e1, e2] is the eigenvector of anelectrical circuit. then imagine a point in a direction of e1, then by using AE = Λ E, it can be said that the A matrix going to be scaled just because of e1, and e2 doesn't have any portion.
Bests
Dear Muhibul Haque Bhuyan
Eigenvalues are a set of scalars associated with a linear system of equations, which are sometimes also known as characteristic roots, latent roots, characteristic values, or proper values. And please go through the attached notes of MIT.
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