Linear Algebra: Advanced Eigen Decomposition Mastery

This quiz provides a comprehensive assessment of your understanding of Eigen Decomposition, a fundamental concept in linear algebra. The questions are designed to be challenging, moving beyond simple procedural calculations to test your deep conceptual and applied knowledge. You will be evaluated on your ability to interpret the geometric meaning of eigenvalues and eigenvectors, understand the conditions for diagonalizability, apply the properties of eigenvalues to analyze matrix transformations, and connect these ideas to real-world applications like Principal Component Analysis (PCA) and dynamical systems.

To succeed, you should be comfortable with the following core concepts and formulas:

• The defining equation of an eigenvector $v$ and eigenvalue $\lambda$:

  $Av = \lambda v$

• The characteristic equation used to find eigenvalues:

  $det(A - \lambda I) = 0$

• The eigen decomposition of a diagonalizable matrix A:

  $A = PDP^{-1}$

  where $P$ is the matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues.

• The computational advantage for matrix powers:

  $A^k = PD^kP^{-1}$

This quiz will rigorously test your mastery of these principles and their far-reaching implications.