eigenvectors AND eigenvalues

description

• eigenvectors: vectors that are stretched or squashed during the transformation
• eigenvalue: the values of stretching or squashing during the transformation, can be negative or not exist xz z

solving

• finding eigenvec and eigenval of matrix A comes down to solving for $\lambda$ (scaling value - eigenvalues) and v (scaling vectors - eigenvectors), in $A\vec{v}=\lambda \vec{v}$
• to solve the eigenvalue $\lambda$, solve: $(A-\lambda I)\vec{v}=0$,
• knowing that we dont want the eigenvectors v = 0, the matrix to solve is $(A-\lambda I)=0$, which looks like this
• fundamental behind (each bullet points are following inferences):
  • equation $(A-\lambda I)\vec{v}=0$ → the product of matrix with nonzero vector v = 0
  • If the linear transformation (A - λI) squishes the space into a lower dimension → the transformation must have a nontrivial null space
    • The null space of the transformation $(A-\lambda I)$ is the set of all vectors v that satisfy the equation $(A-\lambda I)v=0$
    • The equation $(A-\lambda I)v=0$ implies that the linear transformation $(A-\lambda I)$ maps every vector v to the zero vector ⇒ the transformation is not injective ⇒ det = 0 (see injection and determinant relation)
  • it goes down to finding $det(A-\lambda I)=0$
    • eigenvalues: $\lambda$
    • eigenvectors: replace $\lambda$ with the result found → find v1, v2 vectors
    • $V=[v1v2]$
    • S: