--- name: math-thumb-linear-algebra description: "Apply Chapter 7 (Linear Algebra: Choose Stable Matrix Methods Before Computing) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to choose a matrix method from verified structure and report what its residual actually certifies." --- # Linear Algebra: Choose Stable Matrix Methods Before Computing Use this chapter to help the reader make a checked mathematical decision. All 23 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Matrix dimensions and structure; right-hand side; scale and conditioning; precision; memory limit; requested residual and forward accuracy. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Start with sensitivity.** Estimate conditioning and the digit budget before setting solver tolerances. - **Verify claimed structure.** Check relative symmetry, positive definiteness, sparsity pattern, block meaning, and scale before choosing a specialized factorization. - **Choose the direct method by matrix type.** Use Cholesky for verified SPD systems, pivoted LU for general dense systems, and QR or SVD for least squares according to conditioning and rank. - **Avoid unnecessary objects.** Solve for inverse actions, apply Schur complements through block solves, and keep orthogonal factors implicit when possible. - **Budget memory as well as flops.** Consider matrix-free actions, cache blocking, and fill-reducing ordering before the matrix outgrows the machine. - **Verify outputs.** Scale residuals, inspect orthogonality where relevant, and refine a solve only while backward error continues to improve. - **For spectra and compression, inspect cheap evidence first.** Use Gershgorin, eigenvalue ratios, storage counts, and singular-value tails before escalating to shift-invert or large decompositions. - **For Krylov methods, separate problem difficulty from solver design.** Check SPD requirements, conditioning, clustering, preconditioner cost, restart behavior, and true residuals. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Residual and forward error are different quantities. A low-rank storage win, symmetry claim, or converged iteration must pass its own accuracy or structural check. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 7.1.1, 7.2.4, 7.3.4 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C07-D01: Solve a sensitive system and inspect both errors**: rule 7.1.1. - **C07-D02: Fit data and read the residual pattern**: rule 7.2.6. - **C07-D03: Reconstruct a matrix at different ranks**: rule 7.3.5. - **C07-D04: Watch the eigenvalue ratio set the speed of power iteration**: rule 7.3.2. - **C07-D05: See the normal equations square the condition number**: rule 7.2.4. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.