Linear Algebra and Its Applications · Problem 6.2.33

This problem explores how matrix congruence preserves the signature (the count of positive, negative, and zero eigenvalues) of a symmetric matrix. When you conjugate $A$ by $C$ via $C^T A C$, you're performing a change of basis that reshapes $A$ but keeps its fundamental spectral character intact. The key insight is that signature is a congruence invariant—only the magnitudes of eigenvalues change, never their signs. For the chain $C(t)$, you're continuously deforming $C$ toward an orthogonal matrix while preserving nonsingularity; orthogonal matrices are the "signature-neutral" transformations. Finally, the identity matrix has a different signature than $C$ (one is positive-definite, the other indefinite), and you cannot continuously bridge matrices with different signatures without hitting a singular point—a topological obstruction.

This problem explores how matrix congruence preserves the signature (the count of positive, negative, and zero eigenvalues) of a symmetric matrix. When you conjugate $A$ by $C$ via $C^T A C$, you're performing a change of basis that reshapes $A$ but keeps its fundamental spectral character intact. The key insight is that signature is a congruence invariant—only the magnitudes of eigenvalues change, never their signs. For the chain $C(t)$, you're continuously deforming $C$ toward an orthogonal matrix while preserving nonsingularity; orthogonal matrices are the "signature-neutral" transformations. Finally, the identity matrix has a different signature than $C$ (one is positive-definite, the other indefinite), and you cannot continuously bridge matrices with different signatures without hitting a singular point—a topological obstruction.