Abstract
AbstractThe singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, {1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.
Highlights
Inspired by the work [7] on the singular values of involutory matrices some more insight into the singular value decomposition (SVD) of involutory matrices is derived
The singular values σ > of an n × n involutory matrix A appear in pairs (σ, )
The case of singular values σ = is discussed in detail. These singular values may appear in pairs (, ) with closely connected left and right singular vectors or by themselves
Summary
Inspired by the work [7] on the singular values of involutory matrices some more insight into the singular value decomposition (SVD) of involutory matrices is derived. For any matrix A ∈ Cn×n there exists a singular value decomposition (SVD), that is, a decomposition of the form. Where U, V ∈ Cn×n are unitary matrices and Σ ∈ Rn×n is a diagonal matrix with non-negative real numbers on the diagonal. The diagonal entries σi of Σ are the singular values of A. They are ordered such that σ ≥ σ ≥ · · · ≥ σn ≥. A nonsingular matrix A ∈ Cn×n has n positive singular values. The nonuniqueness of the singular vectors mainly depends on the multiplicities of the singular values. Let s > s > · · · > sk > denote the distinct singular values of A with respective k multiplicities θ , θ , .
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