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The 23 pages that link to Singular value decomposition, each with the reason it gives.

• Least squaresRelated: It reveals rank deficiency and supports robust least-squares solutions.

• Jacobian matrixRelated: Its singular values reveal the principal local stretches represented by a Jacobian.

• Linear algebraBroader topic: This factorization reduces a matrix’s action to rotations or reflections and axis-wise scaling.

• Principal component analysisRelated: SVD computes principal directions directly from the data matrix without explicitly forming its covariance matrix.

• RegularizationRelated: Small singular values reveal directions where inversion amplifies noise.

• Jordan normal formCompared with: Unlike Jordan form, it applies to rectangular matrices and is stable under perturbations.

• System of linear equationsRelated: It reveals near-dependencies and supports robust solutions to ill-conditioned systems.

• LU decompositionCompared with: SVD handles rank deficiency and conditioning more robustly, but typically costs more.

• Orthogonal matrixRelated: Orthogonal factors isolate rotations or reflections around the scaling described by singular values.

• Rank (linear algebra)Related: The number of nonzero singular values equals the matrix’s rank.

• Square matrixCompared with: It applies to rectangular matrices where eigenvalue-based analysis may not suffice.

• Eigenvalue problemCompared with: It analyzes arbitrary rectangular matrices where ordinary eigenvalue analysis may not apply.

• QR decompositionCompared with: Unlike QR, it reveals singular values and provides a direct account of numerical rank.

• Spectral theoremCompared with: It applies to arbitrary matrices, whereas the spectral theorem requires structural hypotheses.

• MulticollinearityRelated: Small singular values expose directions in the predictors with little independent information.

• Ridge regressionRelated: In these directions, ridge shrinks unstable estimates according to singular-value size.

• Column spaceRelated: Its left singular vectors describe the column space, including when the matrix is rank-deficient.

• Diagonal matrixRelated: Its central diagonal factor records the independent scale strengths of a linear map.

• Jordan decompositionCompared with: It applies to arbitrary matrices and does not require commuting semisimple and nilpotent summands.

• Matrix theoryBroader topic: It separates a matrix's principal directions from its scaling strengths.

• Fundamental theorem of linear algebraRelated: Its singular vectors expose bases for the fundamental subspaces.

• Eigenvalues and eigenvectorsCompared with: Unlike eigen-decomposition, it applies to rectangular matrices and separates left and right directions.

• Kosambi–Karhunen–Loève theoremCompared with: It gives the finite-dimensional counterpart of optimal orthogonal approximation.