numpy-linalg

Original source / Raw SKILL.md

---
name: numpy-linalg
description: Linear algebra operations in NumPy, including matrix multiplication, SVD, system solving, and least squares fitting. Triggers: linalg, matrix multiplication, SVD, eigenvalues, matrix decomposition, lstsq, multi_dot.
---

## Overview
NumPy's `linalg` module provides high-performance implementations for standard linear algebra routines. It leverages optimized backends (like BLAS/LAPACK) for operations such as Singular Value Decomposition (SVD), batch equation solving, and least-squares optimization.

## When to Use
- Solving systems of linear equations ($Ax = b$).
- Dimensionality reduction or matrix reconstruction using SVD.
- Optimizing chains of matrix multiplications to reduce FLOPs.
- Finding best-fit solutions for overdetermined systems.

## Decision Tree
1. Multiplying two matrices?
   - Use the `@` operator (modern standard).
2. Multiplying 3+ matrices?
   - Use `np.linalg.multi_dot` to optimize multiplication order.
3. Is the matrix square and full-rank?
   - Yes: Use `np.linalg.solve`.
   - No: Use `np.linalg.lstsq` for a best-fit solution.

## Workflows
1. **Solving Multiple Linear Systems in Batch**
   - Organize coefficient matrices into a stack of shape (K, M, M).
   - Organize ordinate values into a stack of shape (K, M).
   - Call `np.linalg.solve(a, b)` to compute all solutions simultaneously.

2. **Rank-Reduced Reconstruction via SVD**
   - Perform Singular Value Decomposition using `np.linalg.svd(a, full_matrices=False)`.
   - Set small singular values in 's' to zero to perform noise reduction.
   - Reconstruct the matrix using `(u * s) @ vh`.

3. **Least Squares Fitting for Overdetermined Systems**
   - Construct the matrix 'A' of variables and vector 'b' of results.
   - Use `np.linalg.lstsq(A, b)` to find the solution that minimizes the Euclidean 2-norm.
   - Retrieve the coefficients and residuals from the returned tuple.

## Non-Obvious Insights
- **Modern Operator:** The `@` operator is preferred over `dot()` for 2D matrix products for readability and intent.
- **Multi-Dot Optimization:** `multi_dot` uses dynamic programming to find the optimal parenthesization of matrix products, which can significantly speed up calculations with varying matrix sizes.
- **Batch Processing:** Most `linalg` functions support "stacked" arrays, processing multiple independent problems in leading dimensions automatically.

## Evidence
- "The @ operator... is preferable to other methods when computing the matrix product between 2d arrays." [Source](https://numpy.org/doc/stable/reference/routines.linalg.html)
- "a must be square and of full-rank... if either is not true, use lstsq for the least-squares best “solution”." [Source](https://numpy.org/doc/stable/reference/generated/numpy.linalg.solve.html)

## Scripts
- `scripts/numpy-linalg_tool.py`: Demonstrates SVD reconstruction and batch solving.
- `scripts/numpy-linalg_tool.js`: Simulated vector normalization logic.

## Dependencies
- `numpy` (Python)

## References
- [references/README.md](references/README.md)