Eigenanalysis

Eigenvector = v such that \(A\mathbf{v} = \lambda \mathbf{v}\), ie the direction of v remains the same but its magnitude changes and \(\lambda\) is the eigenvalue.

• Eigenvectors of a sym. matrix will be orthogonal
• PCA uses the eigenvectors with the largest eigenvalues
• Eigenvalues are roots of the characteristic polynomial where \(\det (A - \lambda I) = 0\)
  • The polynomial is in \(\lambda\) and of degree N, the size of an NxN matrix.
  • You can use SVD to solve arbitrary polynomials by constructing the matrix A, ie: \($\det(\[0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -a & -b & -c & -d\] - \lambda I)\)$
• Eigendecomposition into an orthogonal mat of eigenvectors and diag mat of eigenvalues: \(A = Q\Lambda Q^{-1}\)
  • Extended to non-square matrices with SVD: \(A = U \Sigma V^*\)

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• Eigenvectors of a covmat of vertex positions give you the axes of an OBB!
  • Largest eigenvalue: direction that vertices are most linear
  • Second eigenvalue: spans best-fit plane
  • Third eigenvalue: normal to best-fit plane OBB are way more complex than AABB to check for intersection/collision. But sometimes faster at runtime because you get a much better fit and so less boxes to check!