Matrix Computations

Golub, G. H., & Loan, C. F. V. (2013). Matrix Computations. Johns Hopkins University Press.

Type

Book chapter

Intellectual base

Review or monograph

Year

2013

DOI

10.1137/1.9781421407944

Link

https://doi.org/10.56021/9781421407944

Cited from

economics_finance, mathematics

Cited by

12 citations across 12 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.

• 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.
• 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.

Verification

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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

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