Math for Machine Learning - Chapter 12: Eigen decomposition and Diagonalization

This quiz rigorously tests your advanced understanding of Eigendecomposition and Diagonalization, crucial concepts in linear algebra for machine learning. The questions cover theoretical foundations, computational aspects, and practical implications, designed to challenge even expert learners.

Key formulas and concepts:

• Eigenvalue Equation: For a square matrix $A$, a non-zero vector $v$ is an eigenvector if $Av = \lambda v$, where $\lambda$ is the corresponding eigenvalue.

• Characteristic Equation: Eigenvalues are found by solving $\det(A - \lambda I) = 0$, where $I$ is the identity matrix.

• Diagonalization: A matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$. The columns of $P$ are the linearly independent eigenvectors of $A$, and the diagonal entries of $D$ are the corresponding eigenvalues.

• Conditions for Diagonalization: $A$ is diagonalizable if and only if for every eigenvalue $\lambda$, its algebraic multiplicity equals its geometric multiplicity. A sufficient condition is that $A$ has $n$ distinct eigenvalues. Real symmetric matrices are always orthogonally diagonalizable ($A = QDQ^T$).

• Properties of Eigenvalues:

  • Trace: $\text{tr}(A) = \sum_{i=1}^n \lambda_i$

  • Determinant: $\det(A) = \prod_{i=1}^n \lambda_i$

  • Eigenvalues of $A^k$ are $\lambda^k$.

  • Eigenvalues of $A^{-1}$ are $\lambda^{-1}$ (if $A$ is invertible).

• Matrix Functions: For an analytic function $f(x)$, if $A = PDP^{-1}$, then $f(A) = P f(D) P^{-1}$.

• Geometric Interpretation: Eigendecomposition reveals the invariant directions (eigenvectors) along which a linear transformation acts merely as a scaling (eigenvalues).

• Applications: Fundamental to PCA (Principal Component Analysis) for dimensionality reduction, spectral clustering, solving systems of linear ODEs and recurrence relations, and understanding the stability of dynamical systems.

This quiz demands a deep understanding of these principles and their interconnections. Good luck!