Geometrical and spectral properties of the orthogonal projections of the identity

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 435730, 8pages http://dx.doi.org/10.1155/2013/435730 Research Article Geometrical and Spectral Properties of the Orthogonal Projections of the Identity Luis González, Antonio Suárez, and Dolores García Department of Mathematics, Research Institute SIANI, University of Las Palmas de Gran Canaria, Campus de Tafira, 35017 Las Palmas de Gran Canaria, Spain Correspondence should be addressed to Luis Gonz´ alez; [email protected]c.es Received 28 December 2012; Accepted 10 April 2013 Academic Editor: K. Sivakumar Copyright © 2013 Luis Gonz´ alez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We analyze the best approximation 𝐴𝑁(in the Frobenius sense) to the identity matrix in an arbitrary matrix subspace 𝐴𝑆(𝐴∈R𝑛×𝑛 nonsingular, 𝑆being any fixed subspace of R𝑛×𝑛). Some new geometrical and spectral properties of the orthogonal projection 𝐴𝑁 are derived. In particular, new inequalities for the trace and for the eigenvalues of matrix 𝐴𝑁are presented for the special case that 𝐴𝑁is symmetric and positive definite. 1. Introduction The set of all 𝑛×𝑛real matrices is denoted by R𝑛×𝑛,and 𝐼denotes the identity matrix of order 𝑛.Inthefollowing, 𝐴𝑇and tr(𝐴)denote,asusual,thetransposeandthetrace of matrix 𝐴∈R𝑛×𝑛.Thenotations⟨⋅,⋅⟩𝐹and ‖⋅‖𝐹stand for the Frobenius inner product and matrix norm, defined on the matrix space R𝑛×𝑛. Throughout this paper, the terms orthogonality,angle,andcosinewillbeusedinthesenseof the Frobenius inner product. Our starting point is the linear system 𝐴𝑥=𝑏, 𝐴∈R𝑛×𝑛,𝑥,𝑏∈R𝑛,(1) where 𝐴is a large, nonsingular, and sparse matrix. The resolution of this system is usually performed by iterative methods based on Krylov subspaces (see, e.g., [1,2]). The coefficient matrix 𝐴of the system (1) is often extremely ill-conditioned and highly indefinite, so that in this case, Krylov subspace methods are not competitive without a good preconditioner (see, e.g., [2,3]). Then, to improve the convergence of these Krylov methods, the system (1)can be preconditioned with an adequate nonsingular preconditioning matrix 𝑁, transforming it into any of the equivalent systems 𝑁𝐴𝑥=𝑁𝑏, 𝐴𝑁𝑦=𝑏, 𝑥=𝑁𝑦, (2) the so-called left and right preconditioned systems, respectively. In this paper, we address only the case of the right-hand side preconditioned matrices 𝐴𝑁,butanalogousresultscan be obtained for the left-hand side preconditioned matrices 𝑁𝐴. The preconditioning of the system (1)isoftenperformed in order to get a preconditioned matrix 𝐴𝑁 as close as possible to the identity in some sense, and the preconditioner 𝑁is called an approximate inverse of 𝐴.Theclosenessof𝐴𝑁 to 𝐼maybemeasuredbyusingasuitablematrixnormlike,for instance, the Frobenius norm [4]. In this way, the problem of obtaining the best preconditioner 𝑁(with respect to the Frobenius norm) of the system (1) in an arbitrary subspace 𝑆of R𝑛×𝑛 is equivalent to the minimization problem; see, for example, [5] min 𝑀∈𝑆‖𝐴𝑀−𝐼‖𝐹=‖𝐴𝑁−𝐼‖𝐹.(3) The solution 𝑁to the problem (3) will be referred to as the “optimal” or the “best” approximate inverse of matrix 𝐴 in the subspace 𝑆. Since matrix 𝐴𝑁is the best approximation to the identity in subspace 𝐴𝑆, it will be also referred to as the orthogonal projection of the identity matrix onto the subspace 𝐴𝑆.Althoughmanyoftheresultspresentedinthis paperarealsovalidforthecasethatmatrix𝑁is singular, from now on, we assume that the optimal approximate inverse 𝑁 (and thus also the orthogonal projection 𝐴𝑁) is a nonsingular 2Journal of Applied Mathematics matrix. The solution 𝑁to the problem (3)hasbeenstudied as a natural generalization of the classical Moore-Penrose inverse in [6], where it has been referred to as the 𝑆-MoorePenrose inverse of matrix 𝐴. The main goal of this paper is to derive new geometrical and spectral properties of the best approximations 𝐴𝑁(in the sense of formula (3)) to the identity matrix. Such properties couldbeusedtoanalyzethequalityandtheoreticaleffectiveness of the optimal approximate inverse 𝑁as preconditioner of the system (1). However, it is important to highlight that the purpose of this paper is purely theoretical, and we are not looking for immediate numerical or computational approaches (although our theoretical results could be potentially applied to the preconditioning problem). In particular, the term “optimal (or best) approximate inverse” is used in the sense of formula (3) and not in any other sense of this expression. Among the many different works dealing with practical algorithms that can be used to compute approximate inverses, we refer the reader to for example, [4,7–9]andtothe references therein. In [4], the author presents an exhaustive survey of preconditioning techniques and, in particular, describes several algorithms for computing sparse approximate inverses based on Frobenius norm minimization like, for instance, the well-known SPAI and FSAI algorithms. A different approach (which is also focused on approximate inverses based on minimizing ‖𝐴𝑀−𝐼‖𝐹)canbefound in [7], where an iterative descent-type method is used to approximate each column of the inverse, and the iteration is done with “sparse matrix by sparse vector” operations. When the system matrix is expressed in block-partitioned form, some preconditioning options are explored in [8]. In [9], the idea of “target” matrix is introduced, in the context of sparse approximate inverse preconditioners, and the generalized Frobenius norms ‖𝐵‖2 𝐹,𝐻 =tr(𝐵𝐻𝐵𝑇)(𝐻symmetric positive definite) are used, for minimization purposes, as an alternative to the classical Frobenius norm. The last results of our work are devoted to the special case that matrix 𝐴𝑁is symmetric and positive definite. In this sense, let us recall that the cone of symmetric and positive definite matrices has a rich geometrical structure and, in this context, the angle that any symmetric and positive definite matrix forms with the identity plays a very important role [10]. In this paper, the authors extend this geometrical point of view and analyze the geometrical structure of the subspace of symmetric matrices of order 𝑛,includingthelocationofall orthogonal matrices not only the identity matrix. Thispaperhasbeenorganizedasfollows.InSection2, we present some preliminary results required to make the paper self-contained. Sections 3and 4are devoted to obtain new geometrical and spectral relations, respectively, for the orthogonal projections 𝐴𝑁of the identity matrix. Finally, Section 5closes the paper with its main conclusions. 2. Some Preliminaries Now, we present some preliminary results concerning the orthogonal projection 𝐴𝑁of the identity onto the matrix subspace 𝐴𝑆⊂R𝑛×𝑛. For more details about these results and for their proofs, we refer the reader to [5,6,11]. Taking advantage of the prehilbertian character of the matrix Frobenius norm, the solution 𝑁to the problem (3) canbeobtainedusingtheorthogonalprojectiontheorem. More precisely, the matrix product 𝐴𝑁is the orthogonal projection of the identity onto the subspace 𝐴𝑆, and it satisfies the conditions stated by the following lemmas; see [5,11]. Lemma 1. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, 0≤‖𝐴𝑁‖2 𝐹=tr (𝐴𝑁)≤𝑛, (4) 0≤‖𝐴𝑁−𝐼‖2 𝐹=𝑛−tr (𝐴𝑁)≤𝑛. (5) An explicit formula for matrix 𝑁can be obtained by expressing the orthogonal projection 𝐴𝑁of the identity matrix onto the subspace 𝐴𝑆by its expansion with respect to an orthonormal basis of 𝐴𝑆[5].Thisistheideaofthefollowing lemma. Lemma 2. Let 𝐴∈R𝑛×𝑛 be nonsingular. Let 𝑆be a linear subspace of R𝑛×𝑛 of dimension 𝑑and {𝑀1,...,𝑀𝑑}abasisof 𝑆 such that {𝐴𝑀1,...,𝐴𝑀𝑑}is an orthogonal basis of 𝐴𝑆.Then, the solution 𝑁to the problem (3)is 𝑁=𝑑 ∑ 𝑖=1 tr (𝐴𝑀𝑖) 󵄩󵄩󵄩󵄩𝐴𝑀𝑖󵄩󵄩󵄩󵄩2 𝐹𝑀𝑖,(6) and the minimum (residual)Frobenius norm is ‖𝐴𝑁−𝐼‖2 𝐹=𝑛−𝑑 ∑ 𝑖=1[tr (𝐴𝑀𝑖)]2 󵄩󵄩󵄩󵄩𝐴𝑀𝑖󵄩󵄩󵄩󵄩2 𝐹.(7) Let us mention two possible options, both taken from [5], for choosing in practice the subspace 𝑆and its corresponding basis {𝑀𝑖}𝑑 𝑖=1. The first example consists of considering the subspace 𝑆of 𝑛×𝑛matrices with a prescribed sparsity pattern, that is, 𝑆={𝑀∈R𝑛×𝑛 :𝑚𝑖𝑗 =0∀(𝑖,𝑗)∉𝐾}, 𝐾⊂{1,2,...,𝑛}×{1,2,...,𝑛}.(8) Then, denoting by 𝑀𝑖,𝑗,the𝑛×𝑛matrix whose only nonzero entry is 𝑚𝑖𝑗 =1, a basis of subspace 𝑆is clearly {𝑀𝑖,𝑗 : (𝑖,𝑗) ∈ 𝐾},andthen{𝐴𝑀𝑖,𝑗 : (𝑖,𝑗) ∈ 𝐾}will be a basis of subspace 𝐴𝑆(since we have assumed that matrix 𝐴is nonsingular). In general, this basis of 𝐴𝑆is not orthogonal, so that we only need to use the Gram-Schmidt procedure to obtain an orthogonal basis of 𝐴𝑆, in order to apply the orthogonal expansion (6). For the second example, consider a linearly independent set of 𝑛×𝑛real symmetric matrices {𝑃1,...,𝑃𝑑}and the corresponding subspace 𝑆󸀠=span {𝑃1𝐴𝑇,...,𝑃𝑑𝐴𝑇}, (9) Journal of Applied Mathematics 3 which clearly satisfies 𝑆󸀠⊆{𝑀=𝑃𝐴𝑇:𝑃∈R𝑛×𝑛 𝑃𝑇=𝑃} ={𝑀∈R𝑛×𝑛 :(𝐴𝑀)𝑇=𝐴𝑀}. (10) Hence, we can explicitly obtain the solution 𝑁to the problem (3)forsubspace𝑆󸀠,fromitsbasis{𝑃1𝐴𝑇,...,𝑃𝑑𝐴𝑇}, as follows. If {𝐴𝑃1𝐴𝑇,...,𝐴𝑃𝑑𝐴𝑇}is an orthogonal basis of subspace 𝐴𝑆󸀠, then we just use the orthogonal expansion (6) for obtaining 𝑁.Otherwise,weuseagaintheGram-Schmidt proceduretoobtainanorthogonalbasisofsubspace𝐴𝑆󸀠,and then we apply formula (6). The interest of this second example stands in the possibility of using the conjugate gradient method for solving the preconditioned linear system, when the symmetric matrix 𝐴𝑁is positive definite. For a more detailed exposition of the computational aspects related to these two examples, we refer the reader to [5]. Now, we present some spectral properties of the orthogonal projection 𝐴𝑁. From now on, we denote by {𝜆𝑖}𝑛 𝑖=1 and {𝜎𝑖}𝑛 𝑖=1 the sets of eigenvalues and singular values, respectively, of matrix 𝐴𝑁arranged,asusual,innonincreasingorder,that is, 󵄨󵄨󵄨󵄨𝜆1󵄨󵄨󵄨󵄨≥󵄨󵄨󵄨󵄨𝜆2󵄨󵄨󵄨󵄨≥⋅⋅⋅≥󵄨󵄨󵄨󵄨𝜆𝑛󵄨󵄨󵄨󵄨>0, 𝜎1≥𝜎2≥⋅⋅⋅≥𝜎𝑛>0. (11) The following lemma [11] provides some inequalities involving the eigenvalues and singular values of the preconditioned matrix 𝐴𝑁. Lemma 3. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Then, 𝑛 ∑ 𝑖=1𝜆2 𝑖≤𝑛 ∑ 𝑖=1󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨2≤𝑛 ∑ 𝑖=1𝜎2 𝑖=‖𝐴𝑁‖2 𝐹 =tr (𝐴𝑁)=𝑛 ∑ 𝑖=1𝜆𝑖≤𝑛 ∑ 𝑖=1 󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨≤𝑛 ∑ 𝑖=1𝜎𝑖.(12) The following fact [11]isadirectconsequenceof Lemma 3. Lemma 4. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛. Then, the smallest singular value and the smallest eigenvalue’s modulus of the orthogonal projection 𝐴𝑁 of the identity onto the subspace 𝐴𝑆are never greater than 1. That is, 0<𝜎𝑛≤󵄨󵄨󵄨󵄨𝜆𝑛󵄨󵄨󵄨󵄨≤1. (13) The following theorem [11]establishesatightconnection between the closeness of matrix 𝐴𝑁to the identity matrix and the closeness of 𝜎𝑛(|𝜆𝑛|)totheunity. Theorem 5. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, (1−󵄨󵄨󵄨󵄨𝜆𝑛󵄨󵄨󵄨󵄨)2≤(1−𝜎𝑛)2≤‖𝐴𝑁−𝐼‖2 𝐹 ≤𝑛(1−󵄨󵄨󵄨󵄨𝜆𝑛󵄨󵄨󵄨󵄨2)≤𝑛(1−𝜎2 𝑛). (14) Remark 6. Theorem 5states that the closer the smallest singular value 𝜎𝑛of matrix 𝐴𝑁is to the unity, the closer matrix 𝐴𝑁 willbetotheidentity,thatis,thesmaller ‖𝐴𝑁−𝐼‖𝐹will be, and conversely. The same happens with the smallest eigenvalue’s modulus |𝜆𝑛|of matrix 𝐴𝑁.Inother words, we get a good approximate inverse 𝑁of 𝐴when 𝜎𝑛 (|𝜆𝑛|)issufficientlycloseto1. To finish this section, let us mention that, recently, lower andupperboundsonthenormalizedFrobeniuscondition number of the orthogonal projection 𝐴𝑁of the identity onto the subspace 𝐴𝑆have been derived in [12]. In addition, this work proposes a natural generalization (related to an arbitrary matrix subspace 𝑆of R𝑛×𝑛)ofthenormalized Frobenius condition number of the nonsingular matrix 𝐴. 3. Geometrical Properties In this section, we present some new geometrical properties for matrix 𝐴𝑁,𝑁being the optimal approximate inverse of matrix 𝐴, defined by (3). Our first lemma states some basic properties involving the cosine of the angle between matrix 𝐴𝑁and the identity, that is, cos (𝐴𝑁,𝐼)=⟨𝐴𝑁,𝐼⟩𝐹 ‖𝐴𝑁‖𝐹‖𝐼‖𝐹=tr (𝐴𝑁) ‖𝐴𝑁‖𝐹√𝑛.(15) Lemma 7. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, cos (𝐴𝑁,𝐼)=tr (𝐴𝑁) ‖𝐴𝑁‖𝐹√𝑛=‖𝐴𝑁‖𝐹 √𝑛=√tr (𝐴𝑁) √𝑛,(16) 0≤cos (𝐴𝑁,𝐼)≤1, (17) ‖𝐴𝑁−𝐼‖2 𝐹=𝑛(1−cos2(𝐴𝑁,𝐼)). (18) Proof. First, using (15)and(4)weimmediatelyobtain(16). As a direct consequence of (16), we derive that cos(𝐴𝑁,𝐼)is always nonnegative. Finally, using (5)and(16), we get ‖𝐴𝑁−𝐼‖2 𝐹=𝑛−tr (𝐴𝑁)=𝑛(1−cos2(𝐴𝑁,𝐼))(19) and the proof is concluded. Remark 8. In [13], the authors consider an arbitrary approximate inverse 𝑄of matrix 𝐴and derive the following equality: ‖𝐴𝑄−𝐼‖2 𝐹=(‖𝐴𝑄‖𝐹−‖𝐼‖𝐹)2 +2(1−cos (𝐴𝑄,𝐼))‖𝐴𝑄‖𝐹‖𝐼‖𝐹,(20) 4Journal of Applied Mathematics that is, the typical decomposition (valid in any inner product space) of the strong convergence into the convergence of the norms (‖𝐴𝑄‖𝐹−‖𝐼‖𝐹)2and the weak convergence (1− cos(𝐴𝑄,𝐼))‖𝐴𝑄‖𝐹‖𝐼‖𝐹.Notethatforthespecialcasethat𝑄 is the optimal approximate inverse 𝑁defined by (3), formula (18) has stated that the strong convergence is reduced just to the weak convergence and, indeed, just to the cosine cos(𝐴𝑁,𝐼). Remark 9. More precisely, formula (18)statesthatthecloser cos(𝐴𝑁,𝐼)is to the unity (i.e., the smaller the angle ∠(𝐴𝑁,𝐼) is), the smaller ‖𝐴𝑁−𝐼‖𝐹will be, and conversely. This gives us a new measure of the quality (in the Frobenius sense) of the approximate inverse 𝑁of matrix 𝐴,bycomparingthe minimum residual norm ‖𝐴𝑁−𝐼‖𝐹with the cosine of the angle between 𝐴𝑁and the identity, instead of with tr(𝐴𝑁), ‖𝐴𝑁‖𝐹(Lemma 1), or 𝜎𝑛,|𝜆𝑛|(Theorem 5). So for a fixed nonsingular matrix 𝐴∈R𝑛×𝑛 and for different subspaces 𝑆⊂R𝑛×𝑛,wehave tr (𝐴𝑁)↗𝑛⇐⇒‖𝐴𝑁‖𝐹↗√𝑛⇐⇒𝜎𝑛↗1⇐⇒󵄨󵄨󵄨󵄨𝜆𝑛󵄨󵄨󵄨󵄨↗1 ⇐⇒cos (𝐴𝑁,𝐼)↗1⇐⇒‖𝐴𝑁−𝐼‖𝐹↘0.(21) Obviously, the optimal theoretical situation corresponds to the case tr (𝐴𝑁)=𝑛⇐⇒‖𝐴𝑁‖𝐹=√𝑛⇐⇒𝜎𝑛=1⇐⇒𝜆𝑛=1 ⇐⇒cos (𝐴𝑁,𝐼)=1⇐⇒‖𝐴𝑁−𝐼‖𝐹=0 ⇐⇒𝑁=𝐴−1 ⇐⇒𝐴−1 ∈𝑆. (22) Remark 10. Note that the ratio between cos(𝐴𝑁,𝐼) and cos(𝐴,𝐼)is independent of the order 𝑛of matrix 𝐴. Indeed, assuming that tr(𝐴) =0and using (16), we immediately obtain cos (𝐴𝑁,𝐼) cos (𝐴,𝐼)=‖𝐴𝑁‖𝐹:√𝑛 tr (𝐴):‖𝐴‖𝐹√𝑛=‖𝐴𝑁‖𝐹‖𝐴‖𝐹 tr (𝐴).(23) The following lemma compares the trace and the Frobenius norm of the orthogonal projection 𝐴𝑁. Lemma 11. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, tr (𝐴𝑁)≤‖𝐴𝑁‖𝐹⇐⇒‖𝐴𝑁‖𝐹≤1⇐⇒tr (𝐴𝑁)≤1 ⇐⇒cos (𝐴𝑁,𝐼)≤1 √𝑛,(24) tr (𝐴𝑁)≥‖𝐴𝑁‖𝐹⇐⇒‖𝐴𝑁‖𝐹≥1⇐⇒tr (𝐴𝑁)≥1 ⇐⇒cos (𝐴𝑁,𝐼)≥1 √𝑛.(25) Proof. Using (4), we immediately obtain the four leftmost equivalences. Using (16), we immediately obtain the two rightmost equivalences. The next lemma provides us with a relationship between the Frobenius norms of the inverses of matrices 𝐴and its best approximate inverse 𝑁in subspace 𝑆. Lemma 12. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, 󵄩󵄩󵄩󵄩󵄩𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹󵄩󵄩󵄩󵄩󵄩𝑁−1󵄩󵄩󵄩󵄩󵄩𝐹≥1. (26) Proof. Using (4), we get ‖𝐴𝑁‖𝐹≤√𝑛=‖𝐼‖𝐹=󵄩󵄩󵄩󵄩󵄩(𝐴𝑁)(𝐴𝑁)−1󵄩󵄩󵄩󵄩󵄩𝐹 ≤‖𝐴𝑁‖𝐹󵄩󵄩󵄩󵄩󵄩(𝐴𝑁)−1󵄩󵄩󵄩󵄩󵄩𝐹󳨐⇒ 󵄩󵄩󵄩󵄩󵄩(𝐴𝑁)−1󵄩󵄩󵄩󵄩󵄩𝐹≥1, (27) and hence 󵄩󵄩󵄩󵄩󵄩𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹󵄩󵄩󵄩󵄩󵄩𝑁−1󵄩󵄩󵄩󵄩󵄩𝐹≥󵄩󵄩󵄩󵄩󵄩𝑁−1𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹=󵄩󵄩󵄩󵄩󵄩(𝐴𝑁)−1󵄩󵄩󵄩󵄩󵄩𝐹≥1, (28) and the proof is concluded. The following lemma compares the minimum residual norm ‖𝐴𝑁−𝐼‖𝐹with the distance (with respect to the Frobenius norm) ‖𝐴−1 −𝑁‖𝐹between the inverse of 𝐴and the optimal approximate inverse 𝑁of 𝐴in any subspace 𝑆⊂R𝑛×𝑛. First, note that for any two matrices 𝐴,𝐵∈R𝑛×𝑛 (𝐴nonsingular), from the submultiplicative property of the Frobenius norm, we immediately get ‖𝐴𝐵−𝐼‖2 𝐹=󵄩󵄩󵄩󵄩󵄩𝐴(𝐵−𝐴−1)󵄩󵄩󵄩󵄩󵄩2 𝐹 ≤‖𝐴‖2 𝐹󵄩󵄩󵄩󵄩󵄩𝐵−𝐴−1󵄩󵄩󵄩󵄩󵄩2 𝐹.(29) However, for the special case that 𝐵=𝑁(the solution to the problem (3)), we also get the following inequality. Lemma 13. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, ‖𝐴𝑁−𝐼‖2 𝐹≤‖𝐴‖𝐹󵄩󵄩󵄩󵄩󵄩𝑁−𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹.(30) Proof. Using the Cauchy-Schwarz inequality and (5), we get 󵄨󵄨󵄨󵄨󵄨⟨𝐴−1 −𝑁,𝐴𝑇⟩𝐹󵄨󵄨󵄨󵄨󵄨≤󵄩󵄩󵄩󵄩󵄩𝐴−1 −𝑁󵄩󵄩󵄩󵄩󵄩𝐹󵄩󵄩󵄩󵄩󵄩𝐴𝑇󵄩󵄩󵄩󵄩󵄩𝐹 󳨐⇒ 󵄨󵄨󵄨󵄨󵄨tr ((𝐴−1 −𝑁)𝐴)󵄨󵄨󵄨󵄨󵄨≤󵄩󵄩󵄩󵄩󵄩𝐴−1 −𝑁󵄩󵄩󵄩󵄩󵄩𝐹󵄩󵄩󵄩󵄩󵄩𝐴𝑇󵄩󵄩󵄩󵄩󵄩𝐹 󳨐⇒ 󵄨󵄨󵄨󵄨󵄨tr (𝐴(𝐴−1 −𝑁))󵄨󵄨󵄨󵄨󵄨≤󵄩󵄩󵄩󵄩󵄩𝑁−A−1󵄩󵄩󵄩󵄩󵄩𝐹‖𝐴‖𝐹 󳨐⇒ |tr (𝐼−𝐴𝑁)|≤󵄩󵄩󵄩󵄩󵄩𝑁−𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹‖𝐴‖𝐹 󳨐⇒ 𝑛−tr (𝐴𝑁)≤󵄩󵄩󵄩󵄩󵄩𝑁−𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹‖𝐴‖𝐹 󳨐⇒ ‖𝐴𝑁−𝐼‖2 𝐹≤‖𝐴‖𝐹󵄩󵄩󵄩󵄩󵄩𝑁−𝐴−1󵄩󵄩󵄩󵄩󵄩𝐹. (31) Journal of Applied Mathematics 5 The following extension of the Cauchy-Schwarz inequality, in a real or complex inner product space (𝐻,⟨⋅,⋅⟩),was obtained by Buzano [14]. For all 𝑎,𝑥,𝑏∈𝐻,wehave |⟨𝑎,𝑥⟩⋅⟨𝑥,𝑏⟩|≤1 2(‖𝑎‖‖𝑏‖+|⟨𝑎,𝑏⟩|)‖𝑥‖2.(32) The next lemma provides us with lower and upper bounds on the inner product ⟨𝐴𝑁,𝐵⟩𝐹,forany𝑛×𝑛real matrix 𝐵. Lemma 14. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, for every 𝐵∈R𝑛×𝑛,wehave 󵄨󵄨󵄨󵄨⟨𝐴𝑁,𝐵⟩𝐹󵄨󵄨󵄨󵄨≤1 2(√𝑛‖𝐵‖𝐹+|tr (𝐵)|). (33) Proof. Using (32)for𝑎=𝐼,𝑥=𝐴𝑁,𝑏=𝐵,and(4), we get 󵄨󵄨󵄨󵄨⟨𝐼,𝐴𝑁⟩𝐹⋅⟨𝐴𝑁,𝐵⟩𝐹󵄨󵄨󵄨󵄨≤1 2(‖𝐼‖𝐹‖𝐵‖𝐹+󵄨󵄨󵄨󵄨⟨𝐼,𝐵⟩𝐹󵄨󵄨󵄨󵄨)‖𝐴𝑁‖2 𝐹 󳨐⇒ 󵄨󵄨󵄨󵄨tr (𝐴𝑁)⋅⟨𝐴𝑁,𝐵⟩𝐹󵄨󵄨󵄨󵄨 ≤1 2(√𝑛‖𝐵‖𝐹+|tr (𝐵)|)‖𝐴𝑁‖2 𝐹 󳨐⇒ 󵄨󵄨󵄨󵄨⟨𝐴𝑁,𝐵⟩𝐹󵄨󵄨󵄨󵄨≤1 2(√𝑛‖𝐵‖𝐹+|tr (𝐵)|). (34) The next lemma provides an upper bound on the arithmeticmeanofthesquaresofthe𝑛2terms in the orthogonal projection 𝐴𝑁. By the way, it also provides us with an upper bound on the arithmetic mean of the 𝑛diagonal terms in the orthogonal projection 𝐴𝑁. These upper bounds (valid for any matrix subspace 𝑆) are independent of the optimal approximate inverse 𝑁, and thus they are independent of the subspace 𝑆and only depend on matrix 𝐴. Lemma 15. Let 𝐴∈R𝑛×𝑛 be nonsingular with tr(𝐴) =0and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3).Then, ‖𝐴𝑁‖2 𝐹 𝑛2≤‖𝐴‖2 𝐹 [tr (𝐴)]2, tr (𝐴𝑁) 𝑛≤𝑛‖𝐴‖2 𝐹 [tr (𝐴)]2.(35) Proof. Using (32)for𝑎=𝐴𝑇,𝑥=𝐼,and𝑏=𝐴𝑁,andthe Cauchy-Schwarz inequality for ⟨𝐴𝑇,𝐴𝑁⟩𝐹and (4), we get 󵄨󵄨󵄨󵄨󵄨⟨𝐴𝑇,𝐼⟩𝐹⋅⟨𝐼,𝐴𝑁⟩𝐹󵄨󵄨󵄨󵄨󵄨 ≤1 2(󵄩󵄩󵄩󵄩󵄩𝐴𝑇󵄩󵄩󵄩󵄩󵄩𝐹‖𝐴𝑁‖𝐹+󵄨󵄨󵄨󵄨󵄨⟨𝐴𝑇,𝐴𝑁⟩𝐹󵄨󵄨󵄨󵄨󵄨)‖𝐼‖2 𝐹 󳨐⇒ |tr (𝐴)⋅tr (𝐴𝑁)| ≤𝑛 2(‖𝐴‖𝐹‖𝐴𝑁‖𝐹+󵄨󵄨󵄨󵄨󵄨⟨𝐴𝑇,𝐴𝑁⟩𝐹󵄨󵄨󵄨󵄨󵄨) ≤𝑛 2(‖𝐴‖𝐹‖𝐴𝑁‖𝐹+󵄩󵄩󵄩󵄩󵄩𝐴𝑇󵄩󵄩󵄩󵄩󵄩𝐹‖𝐴𝑁‖𝐹) =𝑛‖𝐴‖𝐹‖𝐴𝑁‖𝐹 󳨐⇒ |tr (𝐴)|‖𝐴𝑁‖2 𝐹≤𝑛‖𝐴‖𝐹‖𝐴𝑁‖𝐹 󳨐⇒ ‖𝐴𝑁‖𝐹 𝑛≤‖𝐴‖𝐹 |tr (𝐴)| 󳨐⇒ ‖𝐴𝑁‖2 𝐹 𝑛2≤‖𝐴‖2 𝐹 [tr (𝐴)]2󳨐⇒ tr (𝐴𝑁) 𝑛≤𝑛‖𝐴‖2 𝐹 [tr (𝐴)]2. (36) Remark 16. Lemma 15 has the following interpretation in terms of the quality of the optimal approximate inverse 𝑁of matrix 𝐴in subspace 𝑆. The closer the ratio 𝑛‖𝐴‖𝐹/|tr(𝐴)| is to zero, the smaller tr(𝐴𝑁)will be, and thus, due to (5), the larger ‖𝐴𝑁−𝐼‖𝐹will be, and this happens for any matrix subspace 𝑆. Remark 17. By the way, from Lemma 15,weobtainthe following inequality for any nonsingular matrix 𝐴∈R𝑛×𝑛. Consider any matrix subspace 𝑆s.t. 𝐴−1 ∈𝑆.Then,𝑁=𝐴−1, and using Lemma 15,weget ‖𝐴𝑁‖2 𝐹 𝑛2=‖𝐼‖2 𝐹 𝑛2=1 n≤‖𝐴‖2 𝐹 [tr (𝐴)]2 󳨐⇒ |tr (𝐴)|≤√𝑛‖𝐴‖𝐹.(37) 4. Spectral Properties In this section, we present some new spectral properties for matrix 𝐴𝑁,𝑁being the optimal approximate inverse of matrix 𝐴, defined by (3). Mainly, we focus on the case that matrix 𝐴𝑁is symmetric and positive definite. This has been motivated by the following reason. When solving a large nonsymmetric linear system (1) by using Krylov methods, a possible strategy consists of searching for an adequate optimal preconditioner 𝑁such that the preconditioned matrix 𝐴𝑁is symmetric positive definite [5]. This enables one to use the conjugate gradient method (CG-method), which is, in general, a computationally efficient method for solving the new preconditioned system [2,15]. Our starting point is Lemma 3,whichhasestablishedthat the sets of eigenvalues and singular values of any orthogonal projection 𝐴𝑁satisfy 𝑛 ∑ 𝑖=1𝜆2 𝑖≤𝑛 ∑ 𝑖=1󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨2≤𝑛 ∑ 𝑖=1𝜎2 𝑖 =𝑛 ∑ 𝑖=1𝜆𝑖≤𝑛 ∑ 𝑖=1 󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨≤𝑛 ∑ 𝑖=1𝜎𝑖.(38) Let us particularize (38)forsomespecialcases. 6Journal of Applied Mathematics First, note that if 𝐴𝑁is normal (i.e., for all 1≤𝑖≤𝑛: 𝜎𝑖=|𝜆𝑖|[16]), then (38)becomes 𝑛 ∑ 𝑖=1𝜆2 𝑖≤𝑛 ∑ 𝑖=1󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨2=𝑛 ∑ 𝑖=1𝜎2 𝑖 =𝑛 ∑ 𝑖=1𝜆𝑖≤𝑛 ∑ 𝑖=1 󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨=𝑛 ∑ 𝑖=1𝜎𝑖.(39) In particular, if 𝐴𝑁is symmetric (𝜎𝑖=|𝜆𝑖|=±𝜆𝑖∈R), then (38)becomes 𝑛 ∑ 𝑖=1𝜆2 𝑖=𝑛 ∑ 𝑖=1󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨2=𝑛 ∑ 𝑖=1𝜎2 𝑖 =𝑛 ∑ 𝑖=1𝜆𝑖≤𝑛 ∑ 𝑖=1 󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨=𝑛 ∑ 𝑖=1𝜎𝑖.(40) In particular, if 𝐴𝑁is symmetric and positive definite (𝜎𝑖= |𝜆𝑖|=𝜆𝑖∈R+), then the equality holds in all (38), that is, 𝑛 ∑ 𝑖=1𝜆2 𝑖=𝑛 ∑ 𝑖=1󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨2=𝑛 ∑ 𝑖=1𝜎2 𝑖 =𝑛 ∑ 𝑖=1𝜆𝑖=𝑛 ∑ 𝑖=1 󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨=𝑛 ∑ 𝑖=1𝜎𝑖.(41) The next lemma compares the traces of matrices 𝐴𝑁and (𝐴𝑁)2. Lemma 18. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Then, (i) for any orthogonal projection 𝐴𝑁 tr ((𝐴𝑁)2)≤‖𝐴𝑁‖2 𝐹=tr (𝐴𝑁),(42) (ii) for any symmetric orthogonal projection 𝐴𝑁 󵄩󵄩󵄩󵄩󵄩(𝐴𝑁)2󵄩󵄩󵄩󵄩󵄩𝐹≤tr ((𝐴𝑁)2)=‖𝐴𝑁‖2 𝐹=tr (𝐴𝑁),(43) (iii) for any symmetric positive definite orthogonal projection 𝐴𝑁 󵄩󵄩󵄩󵄩󵄩(𝐴𝑁)2󵄩󵄩󵄩󵄩󵄩𝐹≤tr ((𝐴𝑁)2)=‖𝐴𝑁‖2 𝐹=tr (𝐴𝑁)≤[tr (𝐴𝑁)]2. (44) Proof. (i) Using (38), we get 𝑛 ∑ 𝑖=1𝜆2 𝑖≤𝑛 ∑ 𝑖=1𝜎2 𝑖=𝑛 ∑ 𝑖=1𝜆𝑖.(45) (ii) It suffices to use the obvious fact that ‖(𝐴𝑁)2‖𝐹≤ ‖𝐴𝑁‖2 𝐹and the following equalities taken from (40): 𝑛 ∑ 𝑖=1𝜆2 𝑖=𝑛 ∑ 𝑖=1𝜎2 𝑖=𝑛 ∑ 𝑖=1𝜆𝑖.(46) (iii) It suffices to use (43) and the fact that (see, e.g., [17, 18]) if 𝑃and 𝑄are symmetric positive definite matrices then tr(𝑃𝑄)≤tr(𝑃)tr(𝑄)for 𝑃=𝑄=𝐴𝑁. The rest of the paper is devoted to obtain new properties about the eigenvalues of the orthogonal projection 𝐴𝑁for the special case that this matrix is symmetric positive definite. First, let us recall that the smallest singular value and the smallest eigenvalue’s modulus of the orthogonal projection 𝐴𝑁are never greater than 1(see Lemma 4). The following theorem establishes the dual result for the largest eigenvalue of matrix 𝐴𝑁(symmetric positive definite). Theorem 19. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Suppose that matrix 𝐴𝑁is symmetric and positive definite. Then, the largest eigenvalue of the orthogonal projection 𝐴𝑁of the identity onto the subspace 𝐴𝑆is never less than 1.Thatis, 𝜎1=𝜆1≥1. (47) Proof. Using (41), we get 𝑛 ∑ 𝑖=1𝜆2 𝑖=𝑛 ∑ 𝑖=1𝜆𝑖󳨐⇒ 𝜆2 𝑛−𝜆𝑛=𝑛−1 ∑ 𝑖=1 (𝜆𝑖−𝜆2 𝑖). (48) Now, since 𝜆𝑛≤1(Lemma 4), then 𝜆2 𝑛−𝜆𝑛≤0. This implies that at least one summand in the rightmost sum in (48)must be less than or equal to zero. Suppose that such summand is the 𝑘th one (1≤𝑘≤𝑛−1). Since 𝐴𝑁is positive definite, then 𝜆𝑘>0,andthus 𝜆𝑘−𝜆2 𝑘≤0󳨐⇒𝜆𝑘≤𝜆2 𝑘󳨐⇒ 𝜆𝑘≥1󳨐⇒𝜆1≥1 (49) and the proof is concluded. In Theorem 19, the assumption that matrix 𝐴𝑁is positive definite is essential for assuring that |𝜆1|≥1,asthefollowing simple counterexample shows. Moreover, from Lemma 4and Theorem 19, respectively, we have that the smallest and largest eigenvalues of 𝐴𝑁(symmetric positive definite) satisfy 𝜆𝑛≤ 1and 𝜆1≥1,respectively.Nothingcanbeassertedabout the remaining eigenvalues of the symmetric positive definite matrix 𝐴𝑁,whichcanbegreaterthan,equalto,orlessthan the unity, as the same counterexample also shows. Example 20. For 𝑛=3,let 𝐴𝑘=(300 0𝑘0 001 ), 𝑘∈R,(50) let 𝐼3be identity matrix of order 3, and let 𝑆be the subspace of all 3×3scalar matrices; that is, 𝑆=span{𝐼3}.Thenthesolution 𝑁𝑘to the problem (3)forsubspace𝑆canbeimmediately obtained by using formula (6) as follows: 𝑁𝑘=tr (𝐴𝑘) 󵄩󵄩󵄩󵄩𝐴𝑘󵄩󵄩󵄩󵄩2 𝐹𝐼3=𝑘+4 𝑘2+10𝐼3(51) andthenweget 𝐴𝑘𝑁𝑘=𝑘+4 𝑘2+10𝐴𝑘=𝑘+4 𝑘2+10(300 0𝑘0 001 ). (52) Journal of Applied Mathematics 7 Let us arrange the eigenvalues and singular values of matrix 𝐴𝑘𝑁𝑘,asusual,innonincreasingorder(asshownin (11)). On one hand, for 𝑘=−2,wehave 𝐴−2𝑁−2 =1 7(300 0−20 001 ), (53) and then 𝜎1=󵄨󵄨󵄨󵄨𝜆1󵄨󵄨󵄨󵄨=3 7 ≥𝜎2=󵄨󵄨󵄨󵄨𝜆2󵄨󵄨󵄨󵄨=2 7 ≥𝜎3=󵄨󵄨󵄨󵄨𝜆3󵄨󵄨󵄨󵄨=1 7. (54) Hence, 𝐴−2𝑁−2 is indefinite and 𝜎1=|𝜆1|=3/7<1. On the other hand, for 1<𝑘<3,wehave(seematrix (52)) 𝜎1=𝜆1=3𝑘+4 𝑘2+10 ≥𝜎2=𝜆2=𝑘𝑘+4 𝑘2+10 ≥𝜎3=𝜆3=𝑘+4 𝑘2+10, (55) and then 𝑘=2:𝜎1=𝜆1=9 7>1, 𝜎2=𝜆2=6 7<1, 𝜎3=𝜆3=3 7<1, 𝑘=5 2:𝜎1=𝜆1=6 5>1, 𝜎2=𝜆2=1, 𝜎3=𝜆3=2 5<1, 𝑘=8 3:𝜎1=𝜆1=90 77>1, 𝜎2=𝜆2=80 77>1, 𝜎3=𝜆3=30 77<1. (56) Hence, for 𝐴𝑘𝑁𝑘positive definite, we have (depending on 𝑘) 𝜆2<1,𝜆2=1,or𝜆2>1. The following corollary improves the lower bound zero on both tr(𝐴𝑁),givenin(4), and cos(𝐴𝑁,𝐼),givenin(17). Corollary 21. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆 be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Suppose that matrix 𝐴𝑁is symmetric and positive definite. Then, 1≤‖𝐴𝑁‖𝐹≤tr (𝐴𝑁)=‖𝐴𝑁‖2 𝐹≤𝑛, (57) cos (𝐴𝑁,𝐼)≥1 √𝑛.(58) Proof. Denote by ‖⋅‖2the spectral norm. Using the wellknown inequality ‖⋅‖2≤‖⋅‖𝐹[19], Theorem 19,and(4), we get ‖𝐴𝑁‖𝐹≥‖𝐴𝑁‖2=𝜎1=𝜆1≥1 󳨐⇒ 1 ≤ ‖𝐴𝑁‖𝐹≤tr (𝐴𝑁)=‖𝐴𝑁‖2 𝐹≤𝑛. (59) Finally, (58)followsimmediatelyfrom(57)and(25). Let us mention that an upper bound on all the eigenvalues moduli and on all singular values of any orthogonal projection 𝐴𝑁can be immediately obtained from (38)and(4)as follows: 𝑛 ∑ 𝑖=1󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨2≤𝑛 ∑ 𝑖=1𝜎2 𝑖=‖𝐴𝑁‖2 𝐹≤𝑛 󳨐⇒ 󵄨󵄨󵄨󵄨𝜆𝑖󵄨󵄨󵄨󵄨,𝜎 𝑖≤√𝑛,∀𝑖=1,2,...,𝑛. (60) Our last theorem improves the upper bound given in (60) for the special case that the orthogonal projection 𝐴𝑁 is symmetric positive definite. Theorem 22. Let 𝐴∈R𝑛×𝑛 be nonsingular and let 𝑆be a linear subspace of R𝑛×𝑛.Let𝑁be the solution to the problem (3). Suppose that matrix 𝐴𝑁is symmetric and positive definite. Then, all the eigenvalues of matrix 𝐴𝑁satisfy 𝜎𝑖=𝜆𝑖≤1+√𝑛 2∀𝑖=1,2,...,𝑛. (61) Proof. First, note that the assertion is obvious for the smallest singular value since |𝜆𝑛|≤1for any orthogonal projection 𝐴𝑁(Lemma 4). For any eigenvalue of 𝐴𝑁, we use the fact that 𝑥−𝑥2≤1/4for all 𝑥>0.Thenfrom(41), we get 𝑛 ∑ 𝑖=1𝜆2 𝑖=𝑛 ∑ 𝑖=1𝜆𝑖 󳨐⇒ 𝜆2 1−𝜆1=𝑛 ∑ 𝑖=2 (𝜆𝑖−𝜆2 𝑖)≤𝑛−1 4 󳨐⇒ 𝜆1≤1+√𝑛 2󳨐⇒ 𝜆𝑖≤1+√𝑛 2 ∀𝑖=1,2,...,𝑛. (62) 5. Conclusion In this paper, we have considered the orthogonal projection 𝐴𝑁(in the Frobenius sense) of the identity matrix onto an 8Journal of Applied Mathematics arbitrary matrix subspace 𝐴𝑆(𝐴∈R𝑛×𝑛 nonsingular, 𝑆⊂ R𝑛×𝑛). Among other geometrical properties of matrix 𝐴𝑁, we have established a strong relation between the quality of the approximation 𝐴𝑁 ≈ 𝐼and the cosine of the angle ∠(𝐴𝑁,𝐼).Also,thedistancebetween𝐴𝑁and the identity has been related to the ratio 𝑛‖𝐴‖𝐹/|tr(𝐴)|(which is independent of the subspace 𝑆). The spectral analysis has provided lower and upper bounds on the largest eigenvalue of the symmetric positive definite orthogonal projections of the identity. Acknowledgments The authors are grateful to the anonymous referee for valuable comments and suggestions, which have improved the earlier versionofthispaper.Thisworkwaspartiallysupportedbythe “Ministerio de Econom´ ıa y Competitividad” (Spanish Government), and FEDER, through Grant Contract CGL201129396-C03-01. References [1] O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, Mass, USA, 1994. [2] Y. Saad, Iterative Methods for Sparse Linear Systems,PWSPublishing, Boston, Mass, USA, 1996. [3] S.-L. Wu, F. Chen, and X.-Q. Niu, “A note on the eigenvalue analysis of the SIMPLE preconditioning for incompressible flow,” Journal of Applied Mathematics,vol.2012,ArticleID 564132, 7 pages, 2012. [4] M. Benzi, “Preconditioning techniques for large linear systems: a survey,” Journal of Computational Physics,vol.182,no.2,pp. 418–477, 2002. [5]G.Montero,L.Gonz ´ alez, E. Fl´ orez, M. D. Garc´ ıa, and A. 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