Decomposition

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.