Books, Papers & Software
Textbooks
| Title | Why it is on the shelf |
|---|---|
| G. Strang, Introduction to Linear Algebra, 6th ed., 2023 | The gentlest complete route: from elimination to the SVD, with unmatched intuition for the four fundamental subspaces. |
| L. N. Trefethen, D. Bau III, Numerical Linear Algebra, SIAM, 1997 | Forty short lectures; still the best bridge from pure theory to floating-point reality. The notes follow its numbering spirit. |
| G. H. Golub, C. F. Van Loan, Matrix Computations, 4th ed., 2013 | The encyclopedia: every algorithm, every flop count, every caveat. |
| R. A. Horn, C. R. Johnson, Matrix Analysis, 2nd ed., 2013 | Where the theorems live: perturbation theory, variational principles, encyclopedic references. |
| S. Axler, Linear Algebra Done Right, 3rd ed., 2015 | The determinant-free exposition; clarifies what needs coordinates and what does not. |
| J. W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997 | Accuracy, conditioning, and roundoff — the analysis layer over the algorithms. |
Landmark papers
| Paper | Contribution |
|---|---|
| C. Eckart, G. Young, Psychometrika 1 (1936) 211–218 | Best low-rank approximation: the truncation theorem every compression method leans on. |
| H. Weyl, Math. Ann. 71 (1912) 441–479 | Stability of the spectrum under perturbation for symmetric operators. |
| M. R. Hestenes, E. Stiefel, J. Res. NBS 49 (1952) 409–436 | Conjugate gradients — the Krylov idea made an algorithm. |
| C. C. Paige, M. A. Saunders, SIAM J. Numer. Anal. 12 (1975) 617–629 | Lanczos for indefinite systems: SYMMLQ and MINRES. |
| Y. Saad, M. H. Schultz, SIAM J. Sci. Stat. Comput. 7 (1986) 856–869 | GMRES for general nonsymmetric systems. |
| N. Halko, P.-G. Martinsson, J. A. Tropp, SIAM Review 53 (2011) 217–288 | The randomized-NLA treatise: sketch-and-solve with full probability guarantees. |
Software
| Project | Role |
|---|---|
| LAPACK / ScaLAPACK | The reference dense direct solvers; everything dense bottoms out here. |
| SuiteSparse | Sparse Cholesky and UMFPACK; where graph theory meets factorizations. |
| PETSc / SLEPc | The portable ecosystems for iterative solvers and eigenproblems at scale. |
| hypre | Home of BoomerAMG — multigrid preconditioning that scales. |
| Eigen | Template C++ linear algebra; the compile-time-fast middle weight. |
| NumPy / SciPy | The interactive workbench; scipy.sparse.linalg wraps many of the above. |
| Julia (LinearAlgebra.jl) | Interactive speed with LAPACK underneath and clean abstractions on top. |
Benchmark matrices
Theory guides, but matrices decide. Two public collections cover most experimental needs: the SuiteSparse Matrix Collection (thousands of real sparse matrices from every application domain, with condition estimates and visualization) and the older Matrix Market set from NIST, still convenient for its generator repertoire (e.g. Wathen, Poisson, orcific test families). When reproducing the timing tables in the research notes, always report the matrix name and collection version alongside the numbers.