Schur classification, factor reuse and singular energy
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Work over the real numbers. State symmetry and strict versus non-strict positivity before invoking a criterion or factorization.
Attempt all three original public proof problems before reading their solutions. Check factors and systems by exact multiplication; recorded drafts are not independently graded proofs.
Loading saved assessment responses…
0 of 3 tasks have recorded responses. No overall mastery score is assigned.
For real t let A(t) have rows (2,1,1),(1,2,2),(1,2,t). Classify every t as positive definite, positive semidefinite but singular, or indefinite. Derive a completed-square block identity and all leading principal minors. At the transition value find the entire kernel and explain why nonnegative leading minors alone would not be an adequate general PSD theorem. For the indefinite range exhibit both a positive and a negative quadratic value rather than relying on a determinant sign alone.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let A have rows (4,2,−2),(2,10,5),(−2,5,6). Construct its lower triangular Cholesky factor L with positive diagonal by the scalar recurrence, checking LLᵀ=A. Reuse that factor to solve Ax=b for b=(−2,2,5) and b=(6,9,2), showing the intermediate forward solutions. For each request determine the minimum and unique minimizer of E_b(x)=½xᵀAx−bᵀx. Prove the minimization through an exact difference identity. Finally, for the trial vector x̂=0 in the second request, derive the energy gap from the residual r=b−Ax̂ and a triangular solve, without forming A⁻¹.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
The real diagonal matrix C=diag(0,−1,2) has all leading principal minors zero. Explain why it is nevertheless not positive semidefinite and identify a principal minor that detects the failure. Now set v=(1,−2,2) and B=vvᵀ. Compute all sizes of principal minors of B and its rank/kernel. For E_b(x)=½xᵀBx−bᵀx, solve and minimize for b=3v, then prove E_b is unbounded below for b=e1 using an explicit null direction. Prove the complete compatibility criterion for a general b in this rank-one example. Explain why neither a zero determinant nor a nonnegative leading-minor list alone settles these minimization questions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For real t let A(t) have rows (2,1,1),(1,2,2),(1,2,t). Classify every t as positive definite, positive semidefinite but singular, or indefinite. Derive a completed-square block identity and all leading principal minors. At the transition value find the entire kernel and explain why nonnegative leading minors alone would not be an adequate general PSD theorem. For the indefinite range exhibit both a positive and a negative quadratic value rather than relying on a determinant sign alone.
Let A have rows (4,2,−2),(2,10,5),(−2,5,6). Construct its lower triangular Cholesky factor L with positive diagonal by the scalar recurrence, checking LLᵀ=A. Reuse that factor to solve Ax=b for b=(−2,2,5) and b=(6,9,2), showing the intermediate forward solutions. For each request determine the minimum and unique minimizer of E_b(x)=½xᵀAx−bᵀx. Prove the minimization through an exact difference identity. Finally, for the trial vector x̂=0 in the second request, derive the energy gap from the residual r=b−Ax̂ and a triangular solve, without forming A⁻¹.
The real diagonal matrix C=diag(0,−1,2) has all leading principal minors zero. Explain why it is nevertheless not positive semidefinite and identify a principal minor that detects the failure. Now set v=(1,−2,2) and B=vvᵀ. Compute all sizes of principal minors of B and its rank/kernel. For E_b(x)=½xᵀBx−bᵀx, solve and minimize for b=3v, then prove E_b is unbounded below for b=e1 using an explicit null direction. Prove the complete compatibility criterion for a general b in this rank-one example. Explain why neither a zero determinant nor a nonnegative leading-minor list alone settles these minimization questions.
