Learn Prove real symmetric positivity criteria, construct Cholesky factors and use exact energy identities to separate unique solutions from singular compatibility and unbounded objectives.
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Positive definiteness requires a strict inequality for every nonzero vector, not just positive diagonal entries.
It is the scalar obtained by applying A to x and taking the dot product with x. Testing coordinate vectors checks the diagonal only; mixed coordinates can reveal a negative direction.
The positivity proof must justify each pivot rather than dividing by an unexamined diagonal entry.
For a nonsingular square matrix, unit-lower LU without row exchanges exists exactly when every leading principal minor is nonzero. Positive symmetric matrices will satisfy stronger positive-minor conditions.
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Schur classification, factor reuse and singular energy
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Derive positive leading-minor and LDL* tests, then prove why semidefinite energy requires all principal minors.
6 practice tasks.
Construct a triangular factor with positive diagonal, solve through it, and identify exactly which energy and least-squares conclusions depend on definiteness.
6 practice tasks.