Eigenvector

Description:

• Vectors that is only scaled after a matrix transformation
  • remain the same direction
• $Ax=\lambda x\to p(\lambda )\doteq det(\lambda I_n-A)=0$
  • where $p$ is the characteristic polynomial, a function of degree $n$
  • $\lambda$ is Eigenvalue denotes the scalar of the eigenvector
    • Can be has multiplicies (repeated root $\lambda$)
  • ex: Ax = 3x then 3 is eigenvalue of x, as it only scale x to 3
• Each distinct eigenvalue ${\lambda }_i$ has a whole subspace ${\varphi }_i\doteq \mathcal{N}({\lambda }_i{I}_n=A)$ of eigenvectors associated to this eigen value
  • called eigenspace
• $A{A}^{⊺}$ or $A^{⊺}A$ have the same positive eigenvalue
• Eigenvalue