What exactly is an eigenvalue?
"To be an eigenfunction, the operator has to reproduce the function with some multiplicative constant."
Is the eigenvalue the multiplicative constant -or- is it the m constant and the function?
Many thanks.
I'll also like to thank Dr. Physics for helping me last time :)
The eigenvalue is the scalar multiplying value. Yes, DrPhysics is a very well known person in the field of physics, and it is a pleasure to have known him, and worked with him, for the past sixteen years. I will relay your thanks.
http://www.drphysics.com/
An eigenvalue is a scalar (constant) that is associated with a specific eigenvector. In the context of linear algebra and matrix theory, an eigenvalue is a scalar λ for which the equation Av = λv holds, where A is a square matrix and v is a nonzero vector. Here's a step-by-step explanation of how to find eigenvalues:
1. Start with a square matrix A.
2. Write down the equation Av = λv, where λ is the eigenvalue to be determined and v is the corresponding eigenvector.
3. Rearrange the equation to the form Av - λv = 0.
4. Factor out v to get (A - λI)v = 0, where I is the identity matrix.
5. Since v is nonzero, the equation (A - λI)v = 0 implies that the matrix (A - λI) must be singular, i.e., it has a determinant of zero.
6. Set up the determinant equation |A - λI| = 0.
7. Solve the determinant equation for λ. The solutions to this equation are the eigenvalues of the matrix A.
Thus, the eigenvalues are the values of λ that satisfy the determinant equation |A - λI| = 0. Each eigenvalue corresponds to a specific eigenvector v, and the eigenvalue is the constant by which the matrix A "scales" the eigenvector v.