 DecompositionBruce C. Dieffenbach ()
Chapter 11 in Conjugate Duality in Economic Analysis, 2026, pp 93-98 from Springer Abstract: Abstract The spectral decomposition of a self-adjoint linear transformation expresses it via its real eigenvalues and orthonormal eigenvectors. The singular-value decomposition of a linear transformation (not necessarily square) is a valuable concept not well appreciated by economists. We derive it as a corollary of the spectral decomposition of a self-adjoint linear transformation. In this book an idempotent is a self-adjoint linear transformation equal to its square. There is a one-to-one correspondence between a subspace and an idempotent having rank equal to the dimension of the subspace. The idempotents are the extreme points of the convex hull of the idempotents of a given rank. To find an optimizing subspace, recast the problem as seeking an optimizing idempotent. One finds the singular-value decomposition of a linear transformation from the spectral decomposition of a Jordan-Wielandt partitioned linear transformation. The singular-value decomposition enables the discovery and computation of relationships among linear transformations. A square linear transformation transforms area by the product of the singular values. Date: 2026 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_11 Ordering information: This item can be ordered from DOI: 10.1007/978-3-032-21396-9_11 Access Statistics for this chapter More chapters in Contributions to Economics from Springer |
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