Matrix Computations, Fourth Edition

Golub, G. (2013). Matrix Computations, Fourth Edition. Johns Hopkins University Press.

Type

Book

Intellectual base

Review or monograph

Year

2013

Pages cited

7-8

DOI

10.56021/9781421407944

ISBN

9781421408590

DOI registrant

Johns Hopkins University Press

Link

https://doi.org/10.56021/9781421407944

Cited from

economics_finance, mathematics

Cited by

15 citations across 15 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Factorization
  • It exposes hidden generative structure: 12 = 2 × 2 × 3 reveals primes invisible in the surface number, a product form inside a joint distribution exposes independence hidden in the joint table, a singular-value decomposition reveals rank structure invisible in the raw matrix.

    This sourceStandard reference for matrix factorizations (LU, QR, Cholesky, eigendecomposition, SVD), each exposing different structure (e.g., rank via SVD) and making a different operation cheap.

• Pivotality
  • In algorithms, pivot elements in Gaussian elimination and pivot-based selection use the term technically.

    This sourceStandard numerical-linear-algebra text establishing 'pivot' as the technical term in Gaussian elimination (the pivot element of a row/column under partial/complete pivoting); the same partition-pivot usage recurs in algorithms texts (Cormen et al., Introduction to Algorithms, for quickselect/quicksort).

• Vector Space
  • The matrix-decomposition toolkit — SVD, eigendecomposition — moved from linear algebra into ML as the engine of recommender systems, dimensionality reduction, and modern transformers.

    This sourceSVD and eigendecomposition — the matrix-decomposition toolkit underlying dimensionality reduction and recommender systems.

Domain-specific

• Bidiagonal matrix
  • An upper or lower bidiagonal matrix has zeros everywhere except possibly on its main diagonal and respectively its first superdiagonal or first subdiagonal; convention should clarify whether permitted entries may themselves be zero.
• Block LU decomposition
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Circulant matrix
  • … and a convenient observable transfers from The circulant generated by c has eigenvalues equal to the DFT of c and shares the Fourier eigenbasis with every other n-by-n circulant. to Periodic convolution is implemented by FFT-transforming a signal and kernel, multiplying spectral components and inverse-transforming..
• Crout matrix decomposition
  • … square or rectangular scope, permutation and pivot rule, lower and upper triangular factors, unit-diagonal convention, update equations, singular cases and numerical stability are explicit. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..
• Hessenberg Matrix
  • This one-band allowance is the narrowest general form to which an arbitrary nonsymmetric matrix can be reduced by stable orthogonal or unitary similarity before iterative eigenvalue computation.
• Matrix
  • These decompositions underlie the majority of numerical algorithms in scientific computing: least-squares solutions (\(QR\)), Gaussian elimination and linear system solving (\(LU\)), principal component analysis and low-rank approximation (\(SVD\)), stability analysis of dynamical systems (eigenvalues), and spectral graph theory (eigenvalues of the adjacency and Laplacian matrices)

    This sourceThe standard numerical-linear-algebra reference, covering LU and general linear systems, QR and full-rank least squares, the singular value decomposition and subspace computations, and unsymmetric, symmetric and large sparse eigenvalue problems; it contains nothing on spectral graph theory or graph Laplacians.

    Supported in partVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Multiply–accumulate operation
  • Golub and Van Loan expose matrix computation as organized families of products and accumulated sums, making the recurrence a stable algorithmic kernel rather than a particular processor instruction.
• Orthogonal Procrustes problem
  • … instance directly demonstrates that the data orientation, centering or scaling, Frobenius objective, orthogonal versus proper-rotation constraint, and rank-deficient nonuniqueness are explicit. to An applied instance preserves the same invariant under a changed notation, scale, implementation, or empirical setting..
• QR Algorithm
  • Under appropriate conditions, repeated steps drive the matrix toward triangular complex Schur form, or quasi-triangular real Schur form.

Mechanisms

• Basis Extraction from a Spanning Set
  • Its honest limitation is that the extracted basis is not unique: it depends on the ordering and the tolerance, so "the basis" is really "a basis this policy produced."

    This sourceShows that numerical basis extraction can select different independent columns according to pivot ordering and the tolerance used to determine rank.

• Pivoted Row Reduction
  • Row reduction has no principled tolerance of its own: the rank verdict then hinges entirely on the zero-threshold, and a small pivot may be a true zero or merely noise — a judgment echelon form cannot make for you.

    This sourceTreats numerical rank as tolerance-dependent and shows why small computed values require a threshold chosen from data accuracy and rounding scale.

• Singular-Value Threshold Scan
  • The scan makes that trade explicit and tunable instead of hidden, and reports how well-conditioned the surviving set is via the ratio of its largest to smallest kept singular value.

    This sourceExplains that numerical rank depends on a chosen singular-value tolerance and that conditioning is read from the ratio of retained extreme singular values.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 1 of 15 citations: 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Links previously used in the corpus

Before the registry existed this work was also linked 4 other ways.

• https://doi.org/10.1137/1.9781421407944 ×1
• https://epubs.siam.org/doi/book/10.1137/1.9781421407944 ×1
• https://jhupbooks.press.jhu.edu/title/matrix-computations ×1
• https://www.press.jhu.edu/books/title/10678/matrix-computations ×1

Registry ID ref:7a709b5bb496 · see in the full table