I think I have an idea of how the Mathematical aspects of Face recognition work. But I am curious as to what an eigenvector would be in this respect. I am trying to understand it through finding out how pixels are encoded: What map takes a pixel into a collection of...
Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue.
However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...
When we are measuring the spin of the electron in the experiment, we choose the spin property as its eigen state for the measurement. The eigen vectors corresponding to these states could be time dependent. Can we still break the problem into solving time independent Schrodinger Equation and...
Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x.
4 4 2
* * *
* * *
Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue?
The Attempt at a Solution
4−λ 4 2
The Cayley-Hamilton Theorem can be used to express the third invariant of the characteristic polynomial obtained from the non-trivial solution of the Eigenvector/Eigenvalue problem. I follow the proof (in Chaves – Notes on Continuum Mechanics) down to the following equation, then get stuck at...
I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using (\sigma _z-\lambda I)X=0,