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The singular values σ &gt; 1 of an  n  ×  n  involutory matrix  A  appear in pairs (σ,1  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.

On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices