Be A(u) = ( u1 * A1 ) + ( u2 * A2 ) + ( u3 * A3 ), u = [ u1 u2 u3 ]T, Ai are known 3x3 real matrices, i=1,2,3.
How to solve the following equation:
- Find the scalar \lambda and the vector u such that
- [ \lambda * I3 - A(u) ] * u = 0,
- I3 is the 3x3 identity matrix, \lambda an eigenvalue of A(u) and u an engenvector of A(u).
Has this problem a tractable and feasible solution?
I would recommend to ask this Q at Mathematics Stack Exchange community too: https://math.stackexchange.com/
You're welcome Jefferson Osowsky! I found it there:
https://math.stackexchange.com/questions/4056685/how-to-find-the-eigenvectors-of-a-matrix-function-that-is-linear-in-its-eigenvec
To me scalar lambda is not possible to find because Det( \lambda * I3 - A(u)) depend on the value of u_i, i=1,2,3. To find eigenvalues matrix A should not depend on unknown u.
In general, the equation [\lambda * I3 - A(u) ] * u = 0] is a system of quadratic equations that can have solution(s) u for certain \lambda.
However, such vector u will not be eigenvector in the true sense. Why?
If u is eigenvector related to an eigenvalue \lambda, then any k-multiple of u is eigenvector.
But in the present case, A(ku)*ku=k2\lambda*u=k*(\lambda*ku), which is not (\lambda*ku).
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