Eigenvectors and eigenvalues

What to learn

  • What is the eigenvectors
  • What is the eigenvalues
  • What is the eigenbasis

    Description

  • After transformation by matrix \(A\), some vectors remain in their own span. thoses are called by eigenvectors. And they have their own eigenvalues the factor is how much they are stretched or squished during transformation.
  • Think about rotation in three dimension. then eigenvector is the axis of rotation. and the eigenvalue is one.
  • As described above, it is expressed in a formular like this. \(A\vec{v} = \lambda I\vec{v}\) , \(\lambda\) is eigenvalue, and \(\vec{v}\) is eigenvector, and \(A\) is transformation.
  • \[(A-\lambda I)\vec{v} = \vec{0}\]
  • We want a nonzero solution for \(\vec{v}\), so matrix \(A-\lambda I\) has to squish the space into lower dimension to make zero vector by nonzero vector.

  • → \(det(A-\lambda I) = 0\)

    \[det\left(\begin{bmatrix} a -\lambda & b \\ c & d-\lambda \end{bmatrix}\right) = (a-\lambda)(d-\lambda) -bc = 0\]

    if any \(\lambda\) makes above fomular, then the eigenvector \(\vec{v}\) will be stratched or squished by \(\lambda\). And if we plug in the \(\lambda\) into matrix, then the eigenvector \(\vec{v}\) will be kernel.

  • \(\lambda\) may exist only one((ex) sheer matrix) or not((ex)rotation matrix).
  • Eigenvector can be multiple or none. if \(2I\) stretch all vector in coordinates. so eigenvector is all vectors.
  • Eigenbasis
    • A set of basis vectors, which are also eigenvertors.
    • Diagonal matrix makes all basis vectors to eigenvectors. and diagonal entries are eigenvalues.
      • Diagonal matrix has powerful computation property.
        • Diagonal matrix \(A = \begin{bmatrix} a& 0 \\ 0 & b \end{bmatrix}\), \(A^{100} = \begin{bmatrix} a^{100}& 0 \\ 0 & b^{100} \end{bmatrix}\)

      • To apply it, when we calculate power of \(B\), we transform the basis,easily find the result.

        \[A^{-1}BA = C, C^{100} = A^{-1}B^{100}A\]

        matrix \(A\) changes basis, and transforms by \(B\), and inverse by \(A^{-1}\).

Next Step

  • Abstract vector spaces

References

- [https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14](https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14)