Every eigenstructure fact developed across Topic 6 came with caveats — eigenvalues might be complex, eigenvectors might fail to span the space, diagonalisation might be impossible. This short, deliberately close-to-the-ground topic states the single most consequential exception in all of applied linear algebra: symmetric matrices never have any of these problems. Their eigenvalues are always real, their eigenvectors (for distinct eigenvalues) are always orthogonal, and — critically for finance — every covariance and correlation matrix is symmetric by construction, which is why the eigenstructure of risk is always this well-behaved case, never the pathological one.
By the end of this topic you should be able to:
- state and understand why a symmetric matrix's eigenvalues are always real;
- state and understand why eigenvectors for distinct eigenvalues of a symmetric matrix are always orthogonal;
- explain, informally, what the spectral theorem promises (full proof deferred to
learn/library/mathematics/linear-algebra-ii/); - work a complete eigenstructure computation end-to-end, using the toolkit built across this whole unit.
In this lesson
- Contents
- The big picture
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