Linear Algebra › Section G · Symmetric matrices and quadratic forms › Topic G.1
Diagonalizing symmetric matrices
Symmetric matrices are the best-behaved square matrices there are: their eigenvalues are always real, and their eigenvectors can always be chosen perpendicular. That makes them easy to diagonalise and is the foundation for quadratic forms, the singular value decomposition and principal component analysis.
In this topic
- Why symmetric matrices deserve their own topic
- Eigenvectors from different eigenvalues are orthogonal
- Orthogonal diagonalisation
- The spectral theorem
- Worked example: a 2 × 2 symmetric matrix
- Worked example: a repeated eigenvalue
- The spectral decomposition
- A preview: the matrix AᵀA
- Common mistakes to avoid
- Key takeaways
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