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On the Singular Value Decomposition Of Spec. Matrices 2020; 8:1–13 Research Article Open Access Heike Faßbender* and Martin Halwaß On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices https://doi.org/10.1515/spma-2020-0001 Received July 30, 2019; accepted November 6, 2019 1 Abstract: The singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, σ ). 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. Keywords: singular value decomposition, (skew-)involutory matrix, (skew-)coninvolutory, consimilarity MSC: 15A23, 65F99 1 Introduction Inspired by the work [7] on the singular values of involutory matrices some more insight into the singular n×n value decomposition (SVD) of involutory matrices is derived. For any matrix A 2 C there exists a singular value decomposition (SVD), that is, a decomposition of the form A = UΣV H , (1.1) n×n n×n where U, V 2 C are unitary matrices and Σ 2 R is a diagonal matrix with non-negative real numbers on the diagonal. The diagonal entries σi of Σ are the singular values of A. Usually, they are ordered such that σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0. The number of nonzero singular values of A is the same as the rank of A. Thus, n×n a nonsingular matrix A 2 C has n positive singular values. The columns uj , j = 1, ... , n of U and the columns vj , j = 1, ... , n of V are the left singular vectors and right singular vectors of A, respectively. From (1.1) we have Avj = σj uj , j = 1, ... , n. Any triplet (u, v, σ) with Av = σu is called a singular triplet of A. In n×n case A 2 R , U and V can be chosen to be real and orthogonal. While the singular values are unique, in general, the singular vectors are not. The nonuniqueness of the singular vectors mainly depends on the multiplicities of the singular values. For simplicity, assume that n×n A 2 C is nonsingular. Let s1 > s2 > ··· > sk > 0 denote the distinct singular values of A with respective Pk H multiplicities θ1, θ2, ... , θk ≥ 1, j=1 θj = n. Let A = UΣV be a given singular value decomposition with Σ = diag(s1Iθ1 , s2Iθ2 , ... , sk Iθk ). Here Ij denotes as usual the j × j identity matrix. Then Ub = U [W1 ⊕ W2 ⊕ ··· ⊕ Wk] and Vb = V [W1 ⊕ W2 ⊕ ··· ⊕ Wk] (1.2) *Corresponding Author: Heike Faßbender: Institut Computational Mathematics/ AG Numerik, TU Braunschweig, Universitätsplatz 2, 38106 Braunschweig, Germany, Email: [email protected] Martin Halwaß: Kampstraße 7, 17121 Loitz, Germany Open Access. © 2020 Heike Faßbender and Martin Halwaß, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 2 Ë Heike Faßbender and Martin Halwaß θj×θj H with unitary matrices Wj 2 C yield another SVD of A, A = UΣb Vb . This describes all possible SVDs of A, see, e.g., [4, Theorem 3.1.1’] or [8, Theorem 4.28]. In case θ = 1, the corresponding left and right singular j p ıαj vector are unique up to multiplication with some e , αj 2 R, where ı = −1. For more information on the SVD see, e.g. [2, 4, 8]. n×n 2 −1 A matrix A 2 C with A = In , or equivalently, A = A is called an involutory matrix. Thus, for any involutory matrix A, A and its inverse A−1 have the same SVD and hence, the same (positive) singular values. H Let A = UΣV be the usual SVD of A with the diagonal of Σ ordered by magnitude, σ1 ≥ σ2 ≥ ... ≥ σn . Noting that an SVD of A−1 is given by A−1 = VΣ−1UH and that the diagonal elements of Σ−1 are ordered by 1 1 1 1 1 1 magnitude, ≥ ≥ ... ≥ , we conclude that σ = , σ = , ... , σn = . That is, the singular values σn σn−1 σ1 1 σn 2 σn−1 σ1 1 of an involutory matrix A are either 1 or pairs (σ, σ ), where σ > 1. This has already been observed in [7] (see Theorem 1 for a quote of the ndings). Here we will describe the SVD A = UΣV H for involutory matrices in 1 detail. In particular, we will note a close relation between the left and right singular vectors of pairs (σ, σ ) of 1 singular values: if (u, v, σ) is a singular triplet, then so is (v, u, σ ). We will observe that some of the singular values σ = 1 may also appear in such pairs. That is, if (u, v, 1) is a singular triplet, then so is (v, u, 1). Other singular values σ = 1 may appear as a single singular triplet, that is (u, ±u, 1). These observations allow to express U as U = VT for a real elementary orthogonal matrix T. With this the SVD of A reads A = VTΣV H . Taking a closer look at the real matrix TΣ we will see that TΣ is an involutory matrix just as A. All relevant information concerning the singular values and the eigenvalues of A can be read o of TΣ. Some of these ndings also follow from [6, Theorem 7.2], see Section 2. n×n 2 As any skew-involutory matrix B 2 C , B = −In can be expressed as B = ıA with an involutory matrix n×n A 2 C , the results for involutory matrices can be transferred easily to skew-involutory ones. n×n We will also consider the SVD of coninvolutory matrices, that is of matrices A 2 C which satisfy AA = In , see, e.g., [5]. As involutory matrices, coninvolutory are nonsingular and have n positive singular values. 1 Moreover, the singular values appear in pairs (σ, σ ) or are 1, see [4]. Similar to the case of involutory matrices, we can give a relation between the matrices U and V in the SVD UΣV of A in the form U = VT. TΣ is a complex coninvolutory matrix consimilar¹ to A and consimilar to the identity. Similar observations have been given in [5]. Some of our ndings also follow from [6, Theorem 7.1], see Section 4. 2n×2n Skew-coninvolutory matrices A 2 C , that is, AA = −I2n , have been studied in [1]. Here we will briey state how with our approach ndings on the SVD of skew-coninvolutory matrices given in [1] can easily be rediscovered. Our goal is to round o the picture drawn in the literature about the singular value decomposition of the four classes of matrices considered here. In particular we would like to make visible the relation between the singular vectors belonging to reciprocal pairs of singular values in the form of the matrix T. The SVD of involutory matrices is treated at length in Section 2. In Section 3 we make immediate use of the results in Section 2 in order to discuss the SVD of skew-involutory matrices. Section 4 deals with coninvolutory matrices. Finally, the SVD of skew-coninvolutory matrices is discussed in Section 5. 2 Involutory matrices n×n 2 −1 Let A 2 C be involutory, that is, A = In holds. Thus, A = A . Symmetric permutation matrices and Hermitian unitary matrices are simple examples of involutory matrices. Nontrivial examples of involutory matrices can be found in [3, Page 165, 166, 170]. The spectrum of any involutory matrix can have at most two n 2 2 elements, λ(A) ⊂ f1, −1g as from Ax = λx for x 2 C nf0g it follows that A x = λAx which gives x = λ x. The SVD of involutory matrices has already been considered in [7]. In particular, the following theorem is given. n×n n×n −1 1 Two matrices A and B 2 C are said to be consimilar if there exists a nonsingular matrix S 2 C such that A = SBS , [5]. On the singular value decomposition Ë 3 n×n 1 1 Theorem 1. Let A 2 C be an involutory matrix. Then B1 = 2 (In −A) and B2 = 2 (In +A) are idempotent, that 2 n n is, Bi = Bi , i = 1, 2. Assume that A has r ≤ 2 eigenvalues −1 and n − r eigenvalues +1 (or r ≤ 2 eigenvalues +1 H H and n − r eigenvalues −1). Then B1 (B2) is of rank r and can be decomposed as B1 = QW (B2 = QW ) where n×r H Q, W 2 C have r linear independent columns. Moreover, W W − I is at least positive semidenite. There are n − 2r singular values of A equal to 1, r singular values which are equal to the eigenvalues of the matrix H 1 H 1 (W W) 2 + (W W − In) 2 and r singular values which are the reciprocal of these. Thus, in [7] it was noted that the singular values of an involutory matrix A may be 1 or may appear in pairs 1 n (σ, σ ). Moreover, the minimal number of singular values 1 is given by 2n−r where r ≤ 2 denotes the number of eigenvalues −1 of A or the number of eigenvalues +1 of A, whichever is smallest.
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