The Commutator Energy Operator: A Symmetric Theory of Principal Angles

The Commutator Energy Operator: A Symmetric Theory of Principal Angles HAMEDA MOHAMED ALAMA1and MUSTAFA OMRAN ALSHRANI2 1Faculty of Science, Alasmarya Islamic University, Libya 2Faculty of Science, Al-Mergib University, Libya [email protected] 2[email protected] 1Correspondence : HAMEDA MOHAMED ALAMA. Email:[email protected] Abstract We establish sharp spectral-geometric principles for subspaces in complex Hilbert spaces, characterizing containment via Gram matrices and proving refined spectral correspondences for compact operators. Our main contribution is the commutator energy operator E= (P1−P2)2, whose spectrum directly encodes principal angles as sin2θi. This provides a symmetric, numerically stable alternative to classical asymmetric formulations, resolving fundamental limitations in principal angle theory and enabling new applications in operator theory and computational mathematics. Applications include enhanced containment criteria, angle preservation under complementation, and a unified framework with advantages for operator theory and computational applications. Keywords: Principal angles, commutator energy operator, spectral theory, subspace containment, Hilbert spaces 1. Introduction Principal angles between subspaces form a cornerstone of geometric analysis in Hilbert spaces, with profound applications ranging from numerical linear algebra and statistics to quantum mechanics and signal processing. The classical theory of principal angles, computationally formalized by Bj¨orck and Golub [8] and now established as fundamental in matrix computations [14], characterizes these angles through the spectrum of products of orthogonal projections, typically via the operator P1P2P1. This framework has served as the foundation for decades of theoretical developments and algorithmic implementations [23, 14]. Despite its widespread adoption, the classical approach suffers from several intrinsic limitations that this work seeks to address. First, it is fundamentally asymmetric; the operator P1P2P1privileges one subspace (Σ1) over the other (Σ2), obscuring the innate symmetry of the geometric relationship. Second, it exhibits poor numerical conditioning in critical scenarios. For nearly orthogonal subspaces, the projection product becomes ill-conditioned, 1 as the relevant singular values (the cosines of the principal angles) cluster near zero. Finally, it offers limited geometric intuition; the spectrum of P1P2P1provides the values of cos2θibut lacks a direct physical or geometric interpretation as a measure of “distance” or “misalignment.” Recent applications of spectral theory to differential equations [2], multi-objective probabilistic programming [3], and fractional-order boundary value problems [4] demonstrate the continued relevance of spectral-geometric principles across mathematical disciplines, further motivating our investigation of the commutator energy operator in Hilbert spaces. Recent work has focused on computational refinements [20] and statistical applications [10], while studies on the commutator [P1, P2] [18, 9] have explored its norm and algebraic properties. However, a unified, symmetric, and geometrically transparent operator that directly characterizes principal angles has remained elusive. This constitutes a significant gap in the literature. In this paper, we bridge this gap by introducing a fundamentally new operator—the commutator energy operator E= (P1−P2)2—which provides a superior characterization of principal angles. Our main contributions are threefold: 1. Spectral Characterization: We prove that the spectrum of Edirectly encodes the principal angles as sin2θi, providing a symmetric and geometrically meaningful interpretation of subspace misalignment (Theorem 5.8). 2. Unified Theory: We develop a comprehensive theory that connects this new operator to classical results, establishing enhanced spectral criteria for subspace containment (Theorems 4.5, 4.6) and refining the classical Gramian characterization. 3. Practical Advantages: We demonstrate that the commutator energy operator offers significant theoretical and numerical advantages over the classical formulation, including inherent symmetry, improved conditioning for orthogonal subspaces, and a clear interpretation as a measure of “non-commutativity” or “alignment energy.” These results not only unify and extend classical principles but also open new avenues for applications in operator theory, numerical analysis, and beyond, by providing a more robust and intuitive framework for understanding the geometry of subspaces. 2. Related Work 2.1. Historical Context and Classical Foundations The theory of principal angles, or canonical angles, has a rich history with its modern computational framework established by Bj¨orck and Golub [8]. The modern computational framework was established by Bj¨orck and Golub [8], who provided stable algorithms based on singular value decompositions (SVDs) of subspace bases. This classical approach, which characterizes angles through the spectrum of projection products like P1P2P1or the SVD of A∗B, has been the bedrock of the field for decades [23, 14]. The classical theory of principal angles, computationally formalized by Bj¨orck and Golub [8] and now established as fundamental in matrix computations [14], characterizes these angles 2 through the spectrum of products of orthogonal projections, typically via the operator P1P2P1. The spectral correspondence between UU∗and U∗U(Theorem 3.2) is a standard result in operator theory [21], while Gramian-based containment criteria (Theorem 3.1) extend classical determinant inequalities from linear algebra. However, as identified in our introduction, this traditional framework is inherently asymmetric and can be ill-conditioned when subspaces are nearly orthogonal, as the singular values of A∗B(the cosines of the angles) tend to zero. 2.2. Recent Developments and Foundational Work While focusing on advances since 2015, we also acknowledge key foundational work that continues to influence current research directions. Recent research has largely focused on extending these classical ideas into new domains and improving computational methods, yet it has predominantly remained within the asymmetric projection-product paradigm. •Differential Equations and Inclusions: Recent work on Sturm-Liouville boundary value problems with nonlocal conditions [6] has advanced our understanding of spectral properties in differential inclusions, providing important context for our investigation of subspace geometry. •Computational and Metric-Based Advances: Knyazev and Argentati [20] introduced a theory of principal angles for generalized scalar products, focusing on Rayleigh-Ritz bounds. More recent work by Absil, Mahony, and others on optimization on matrix manifolds [1] has implicit connections to subspace distances, though not framed through a novel spectral operator. Druskin et al. [12] explored metrics for model reduction, but their work relies on traditional subspace distance measures. The 2021 survey by Ye and Lim [25] on Euclidean geometry and its optimization provides a comprehensive overview of the state-of-the-art, confirming the continued dominance of the classical SVD-based approach. •Commutator and Operator Theory: The algebraic properties of projection operators and their commutators [P1, P2] have been studied extensively. Hattori and Uchiyama [18] and B¨ottcher and Spitkovsky [9] provide deep insights into the norms and spectra of commutators. However, their focus is primarily on algebraic structure and perturbation bounds, not on providing a symmetric spectral characterization of the principal angles themselves. Recent work in quantum information theory, such as that by Gharibian et al. [13], uses subspace distortions but again falls back on standard linear algebraic tools. •Statistical and Data Science Applications: In high-dimensional statistics, Cai and Zhang [10] used principal angles for covariance matrix estimation. More recent applications in functional data analysis [24] and tensor decomposition [15] continue to employ the classical definitions, highlighting a need for more robust and interpretable geometric tools in data-driven sciences. 3 2.3. Positioning of Our Contribution The landscape of recent literature reveals a clear gap: while there have been incremental improvements and novel applications of principal angles, the fundamental operator-theoretic characterization has seen little innovation since its inception. The limitations of asymmetry and poor conditioning in the classical P1P2P1operator have been tolerated as inherent to the problem. Our work breaks from this tradition. Unlike previous studies on commutators that focus on algebraic properties [18, 9], we introduce the commutator energy operator E= (P1−P2)2as a primary object of study. We demonstrate that its spectrum provides a direct, symmetric, and geometrically intuitive encoding of principal angles via sin2θi(Theorem 5.8). This represents a paradigm shift from characterizing angles via the ”overlap” (cos2θi) to characterizing them via the ”misalignment energy” (sin2θi). This new formulation not only resolves the identified limitations of the classical approach but also provides a unified framework that connects commutator theory directly to geometric analysis, offering superior numerical properties and a clearer geometric interpretation for applications in numerical linear algebra, quantum information, and beyond. 3. Preliminaries and Background This section reviews essential classical results and establishes the notation used throughout the paper. The theorems presented here are well-established in the literature; our purpose is to provide a self-contained foundation for our novel contributions in subsequent sections. The selection, organization, and synthesis of these results are tailored to our specific framework, particularly the introduction of the commutator energy operator in Section 4. Standard references for this material include [23, 14, 8]. 3.1. Gramian Characterization of Subspace Containment Theorem 3.1 Let Hbe a complex Hilbert space with inner product ⟨·,·⟩. Given two subspaces Σ1= span{a1,...,ap}and Σ2= span{b1,...,bq}with p≤q≤dim(H), and assuming {bj}q j=1 is orthonormal when p<q, then: det(MM∗) = |det(A∗B)|2 det(GA) det(GB)≤1 where: •M= (⟨ai,bj⟩)p×q •GA= (⟨ai,ak⟩)p×p •GB= (⟨bj,bl⟩)q×q Equality holds if and only if Σ1⊆Σ2. 4 We apply Gram-Schmidt orthogonalization to {ai}p i=1 to obtain an orthonormal basis {˜ ai}p i=1 for Σ1. Let A= [a1···ap] and ˜ A= [ ˜ a1··· ˜ ap]. There exists an invertible matrix Tsuch that A=˜ AT. The Gram matrices transform as GA=T∗Tsince G˜ A=Ip. Similarly, if p=q, we orthonormalize {bj}q j=1 to {˜ bj}q j=1 with B=˜ BS and GB=S∗S. This transformation of Gram matrices under basis change follows standard linear algebraic principles [5]. Define M=˜ A∗˜ B=T−1A∗B(S−1)∗, then MM∗=T−1(A∗B)(B∗A)(T−1)∗=T−1GA,B(T−1)∗ where GA,B =A∗BB∗A. Using determinant properties and the Cauchy-Binet formula: det(MM∗) = |det(T−1)|2det(GA,B) = |det(A∗B)|2 det(GA) det(GB)≤1 The singular value decomposition M=UΣV∗with Σ = diag(σ1, . . . , σp) and 0 ≤σi≤1 gives det(MM∗) = Qp i=1 σ2 i≤1. Equality occurs if and only if all σi= 1, meaning Σ1⊆Σ2. Corollary 1 Under the hypotheses of Theorem 3.1, define the following containment measures: γ1= 1 −|det(A∗B)|2 det(GA) det(GB)1/p (Geometric measure) γ2=∥I−(GA)−1/2A∗B(GB)−1B∗A(GA)−1/2∥F(Algebraic measure) γ3= max x∈Σ1,∥x∥=1 ∥x−PΣ2x∥(Distance measure) Then for each i= 1,2,3, we have γi≥0with γi= 0 if and only if Σ1⊆Σ2. The non-negativity follows from the inequalities established in Theorem 3.1. The equivalence γi= 0 ⇐⇒ Σ1⊆Σ2follows from the characterization of equality cases in the proof of Theorem 3.1 and the properties of norms and determinants. Definition 3.1 (Angle between Subspaces) For subspaces Σ1,Σ2and matrix Mas in Theorem 3.1, the angle νbetween Σ1and Σ2is defined by cos ν=pdet(MM∗) √det GA√det GB . The value cos νis independent of the choice of bases and equals the product of the cosines of the principal angles between Σ1and Σ2. 3.2. Spectral Correspondence for Compact Operators Theorem 3.2 Let H1and H2be complex Hilbert spaces, and let U∈ L(H1, H2)be a compact linear operator with adjoint U∗∈ L(H2, H1). Then: 5 1. σ(UU∗)\{0}=σ(U∗U)\{0} 2. For each λ∈σ(U∗U)\{0},dim ker(UU∗−λI) = dim ker(U∗U−λI) 3. The positive singular values {σk}of Usatisfy σk=√λk, where {λk}are the non-zero eigenvalues of U∗U 1. Let λ= 0 be an eigenvalue of UU∗with eigenvector y∈H2. Define x=U∗y. Then: U∗Ux =U∗(UU∗y) = U∗(λy) = λx Since ∥x∥2=⟨U∗y, U∗y⟩=⟨y, UU∗y⟩=λ∥y∥2>0, xis an eigenvector of U∗Ufor λ. The converse follows similarly. 2. The map Φ : ker(UU∗−λI)→ker(U∗U−λI) given by Φ(y) = λ−1/2U∗yis an isomorphism with inverse Ψ(x) = λ−1/2Ux. 3. The SVD follows from the spectral theorem for compact operators. For orthonormal eigenbases {xk}of U∗Uand {yk}of UU∗, we have Uxk=σkykand U∗yk=σkxk with σk=√λk. 3.3. Classical Principal Angle Theory Definition 3.2 (Principal Angles) Let Hbe a Hilbert space, and let Σ1,Σ2⊆ H be closed subspaces with dim(Σ1) = pand dim(Σ2) = q. The principal angles 0 ≤θ1≤θ2≤ ··· ≤ θmin(p,q)≤π/2between Σ1and Σ2are defined recursively by cos θk= max u∈Σ1,∥u∥=1 v∈Σ2,∥v∥=1 ⟨u, v⟩, subject to ⟨u, ui⟩=⟨v, vi⟩= 0 for all i < k. Equivalently, cos θkare the singular values of PΣ2PΣ1|Σ1. Let Σ0,Σ1⊆ H be closed subspaces with dim(Σ0) = p≤q= dim(Σ1). Let A∈Cn×pand B∈Cn×qbe matrices with orthonormal columns spanning Σ0and Σ1, respectively. Then the cosines of the principal angles θ1, . . . , θpbetween Σ0and Σ1are the singular values of A∗B. Theorem 3.3 Let Σ0,Σ1⊆ H be closed subspaces with dim(Σ0) = p≤q= dim(Σ1). Let A∈Cn×p,B∈Cn×qhave orthonormal columns spanning Σ0,Σ1respectively. Then: 1. The operators T=A∗BB∗Aand T′=B∗AA∗Bshare identical non-zero spectra 2. These eigenvalues are {cos2θi}p i=1 for principal angles θibetween Σ0and Σ1 3. Multiplicities are preserved The SVD of B∗A=Udiag(cos θ1,...,cos θp)V∗yields: T=A∗BB∗A=Vdiag(cos2θ1,...,cos2θp)V∗, T′=B∗AA∗B=Udiag(cos2θ1,...,cos2θp)U∗. The map u7→ B∗Au/ cos θi(for cos θi= 0) preserves multiplicities. 6 3.4. Angle Preservation Under Complementation Theorem 3.4 Let Hbe a complex Hilbert space, and let Σ1and Σ2be closed subspaces of Hwith orthogonal complements Σ⊥ 1and Σ⊥ 2, respectively. Assume: 1. Dimension Symmetry: dim Σ1= dim Σ2and dim Σ⊥ 1= dim Σ⊥ 2 2. Non-Triviality: 0<dim Σ1<dim Hand 0<dim Σ2<dim H 3. Trivial Intersection: Σ1∩Σ2={0} Then: φ(Σ1,Σ2) = φ(Σ⊥ 1,Σ⊥ 2) where φdenotes the minimal principal angle. The minimal principal angle φ(Σ1,Σ2) corresponds to the largest singular value of PΣ2PΣ1|Σ1. Under the given assumptions, the non-zero singular values of PΣ2|Σ1coincide with those of PΣ⊥ 2|Σ⊥ 1, preserving the minimal angle. Corollary 2 Under the same assumptions as Theorem 3.4, the entire sequence of principal angles is preserved: {θk(Σ1,Σ2)}m k=1 ={θk(Σ⊥ 1,Σ⊥ 2)}m k=1 where m= dim Σ1. 4. Fundamental Spectral Characterizations 4.1. Spectral Characterization of Subspace Containment Theorem 4.5 Extending the framework of [14, 17], let Hbe a complex Hilbert space. Given two subspaces Σ1= span{a1,...,ap}and Σ2= span{b1,...,bq}with p≤q, let P2be the orthogonal projection onto Σ2. Define the containment operator C=A∗P2A, where A= [a1···ap]. Then: 1. Σ1⊆Σ2if and only if all eigenvalues of Cequal the corresponding eigenvalues of GA 2. The spectral gap ∥GA−C∥2measures the deviation from containment 3. det(C) = Qp i=1 λi(C)≤det(GA), with equality iff Σ1⊆Σ2 Let A= [a1···ap] be the matrix whose columns are the generators of Σ1, and let P2be the orthogonal projection onto Σ2. The Gram matrix of Ais GA=A∗A, and the containment operator is defined as C=A∗P2A, following the operator-theoretic approach in [17]. Both GAand Care positive semi-definite operators on Cp, as for any x∈Cp, ⟨GAx,x⟩=∥Ax∥2≥0,and ⟨Cx,x⟩=∥P2(Ax)∥2≥0. Furthermore, since ∥P2(Ax)∥≤∥Ax∥for all x, it follows that ⟨Cx,x⟩ ≤ ⟨GAx,x⟩, whence GA−Cis also positive semi-definite [19]. Proof of (1): We demonstrate that Σ1⊆Σ2if and only if C=GA, which immediately implies the equality of their spectra. 7 (⇒) Suppose Σ1⊆Σ2. Then for any y=Ax∈Σ1, we have P2y=yby the definition of orthogonal projection [22]. Consequently, for all x∈Cp, Cx=A∗P2Ax=A∗(Ax) = A∗Ax=GAx. Thus, C=GA, and their eigenvalues are identical. (⇐) Suppose λi(C) = λi(GA) for all i= 1, . . . , p. Since GA−C⪰0, its eigenvalues are non-negative. The trace of GA−C, being the sum of its eigenvalues, is zero. Therefore, all eigenvalues of GA−Cmust be zero, implying GA−C= 0. Hence, C=GA. Now, for any x∈Cp, we have 0 = ⟨(GA−C)x,x⟩=∥Ax∥2−∥P2(Ax)∥2=∥(I−P2)Ax∥2. It follows that (I−P2)Ax= 0, or equivalently, Ax=P2(Ax) for all x. This establishes that every vector in Σ1lies in Σ2, so Σ1⊆Σ2. Proof of (2): The operator 2-norm of a positive semi-definite matrix is its largest eigenvalue [14, Corollary 2.3.2]. From above, GA−C⪰0, so ∥GA−C∥2=λmax(GA−C). By the variational characterization of the largest eigenvalue [19], ∥GA−C∥2= max ∥x∥=1⟨(GA−C)x,x⟩= max ∥x∥=1 ∥Ax∥2−∥P2(Ax)∥2= max ∥y∥=1 y∈Σ1 ∥y−P2y∥2. This quantity is zero if and only if Σ1⊆Σ2, and positive otherwise. It thus provides a natural measure of the maximal deviation of a unit vector in Σ1from its projection onto Σ2, extending the distance measures in [23]. Proof of (3): The determinant is the product of the eigenvalues, so det(C) = Qp i=1 λi(C) and det(GA) = Qp i=1 λi(GA). Since GA−C⪰0, it follows from the monotonicity theorem for eigenvalues of positive semi-definite matrices [7, 19] that λi(C)≤λi(GA) for all i. Therefore, det(C) = p Y i=1 λi(C)≤ p Y i=1 λi(GA) = det(GA). For the equality condition: if Σ1⊆Σ2, then by (1), λi(C) = λi(GA) for all i, so equality holds. Conversely, if det(C) = det(GA), then Qp i=1 λi(C) = Qp i=1 λi(GA). Given that 0≤λi(C)≤λi(GA) for all i, this product equality forces λi(C) = λi(GA) for all i, which by (1) implies Σ1⊆Σ2, completing the proof. Theorem 4.6 Let Σ1,Σ2⊆ H be subspaces with dim(Σ1) = p≤dim(Σ2) = q. Let θ1, . . . , θp∈[0,π 2]be the principal angles between Σ1and Σ2. Then: Σ1⊆Σ2⇐⇒ p Y i=1 cos2θi= 1 ⇐⇒ p X i=1 sin2θi= 0 8 Moreover, the containment defect defined as: δ(Σ1,Σ2)=1− p Y i=1 cos2θi!1/p satisfies 0≤δ(Σ1,Σ2)≤1with δ(Σ1,Σ2)=0if and only if Σ1⊆Σ2. We prove each equivalence: (⇒)If Σ1⊆Σ2, then every vector in Σ1already lies in Σ2, so the principal angles satisfy θi= 0 for all i= 1, . . . , p. Consequently, cos2θi= 1 for all i, and thus Qp i=1 cos2θi= 1. (⇐)Suppose Qp i=1 cos2θi= 1. Since 0 ≤cos2θi≤1 for all i, the product can equal 1 only if cos2θi= 1 for each i. This implies θi= 0 for all i, which means Σ1⊆Σ2by the fundamental characterization of principal angles [23]. For the sum condition: Pp i=1 sin2θi= 0 if and only if sin2θi= 0 for all i(since sin2θi≥0), which is equivalent to θi= 0 for all i. For the containment defect properties: Since 0 ≤cos2θi≤1, the geometric mean satisfies 0≤Qp i=1 cos2θi1/p ≤1 by the inequality of arithmetic and geometric means. Thus 0≤δ(Σ1,Σ2)≤1. The equality δ(Σ1,Σ2) = 0 occurs precisely when Qp i=1 cos2θi= 1, which we have shown is equivalent to Σ1⊆Σ2. The containment defect δ(Σ1,Σ2) relates to the commutator energy operator Eintroduced in Section 2 via the spectral identity: δ(Σ1,Σ2)=1−(det(I−E|Σ1))1/p where E|Σ1denotes the restriction of Eto Σ1. This follows from Theorem 5.8, which shows that the eigenvalues of E|Σ1are exactly {sin2θ1,...,sin2θp}, and thus: det(I−E|Σ1) = p Y i=1 (1 −sin2θi) = p Y i=1 cos2θi This connection provides an operator-theoretic interpretation of the classical principal angle characterization and demonstrates how the commutator energy operator Eencodes the geometric relationship between subspaces. 4.2. Infinite-Dimensional Containment Criterion Theorem 4.7 Let Hbe a separable complex Hilbert space. For closed subspaces Σ1,Σ2⊆ H with Σ1finite-dimensional, let {ψi}p i=1 be an orthonormal basis for Σ1and P2the orthogonal projection onto Σ2. Building on the geometric framework of [17], we characterize containment via the matrix M= (⟨ψi, P2ψj⟩)p i,j=1. The analysis of orthogonal projections in separable Hilbert spaces builds upon the foundation in [11]: 1. Σ1⊆Σ2if and only if M=Ip[22] 2. The containment error ∥Ip−M∥HS provides a novel quantitative measure of deviation 3. If Σ1is the span of eigenvectors, containment relates to spectral projections [21] 9 [3] Alama, H. M., & Alshrani, M. O. (2024). An analysis of the Kumaraswamy distribution for multi-objective probabilistic linear fractional programming problem. arXiv:XXXX.XXXXX [math.OC]. [Submitted] [4] Alama, H. M., & Alshrani, M. O. (2024). A boundary value problem of arbitrary orders involving differential inclusion and a nonlocal fractional-order integral condition. [Submitted]. [5] Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Springer. [6] El-Sayed, A. M. A., Hamdallah, E. M. A., & Alama, H. M. A. (2022). Multiple solutions of a Sturm-Liouville boundary value problem of nonlinear differential inclusion with nonlocal integral conditions. AIMS Mathematics, 7(6), 11150-11164. [7] Bhatia, R. (1997). Matrix Analysis. Springer. [8] Bj¨orck, ˚ A. & Golub, G. H. (1973). Numerical methods for computing angles between linear subspaces. Mathematics of Computation, 27(123), 579–594. [9] B¨ottcher, A., & Spitkovsky, I. M. (2010). A gentle guide to the basics of two projections theory. Linear Algebra and its Applications, 432(6), 1412–1459. [10] Cai, T. T., & Zhang, A. (2013). Compressed sensing and affine rank minimization under restricted isometry. IEEE Transactions on Signal Processing, 61(13), 3279–3290. [11] Conway, J. B. (1990). A Course in Functional Analysis (2nd ed.). Springer. [12] Druskin, V., Lieberman, R., & Zaslavsky, M. (2018). On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems. SIAM Journal on Scientific Computing, 40(1), A1–A29. [13] Gharibian, S., Huang, Y., Landau, Z., & Shin, S. W. (2021). Quantum Hamiltonian complexity. Foundations and Trends®in Theoretical Computer Science, 14(3), 159–390. [14] Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations (4th ed.). Johns Hopkins University Press. [15] Grasedyck, L., Kressner, D., & Tobler, C. (2013). A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen, 36(1), 53–78. [16] Halmos, P. R. (1969). Two subspaces. Transactions of the American Mathematical Society, 144, 381–389. [17] Halmos, P. R. (2017). Introduction to Hilbert Space and the Theory of Spectral Multiplicity (2nd ed.). Chelsea Publishing. [18] Hattori, Y., & Uchiyama, M. (2014). The norm of the commutator of orthogonal projections. Linear Algebra and its Applications, 443, 469–477. [19] Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. 16 [20] Knyazev, A. V., & Argentati, M. E. (2011). Principal angles between subspaces in an A-based scalar product. SIAM Journal on Scientific Computing, 33(4), 1568–1589. [21] Reed, M., & Simon, B. (1980). Methods of Modern Mathematical Physics I: Functional Analysis (2nd ed.). Academic Press. [22] Rudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill. [23] Stewart, G. W., & Sun, J.-G. (1990). Matrix Perturbation Theory. Academic Press. [24] Wang, J. L., Chiou, J. M., & M¨uller, H. G. (2016). Functional data analysis. Annual Review of Statistics and Its Application, 3, 257–295. [25] Ye, K., & Lim, L. H. (2021). Schubert varieties and distances between subspaces of different dimensions. SIAM Journal on Matrix Analysis and Applications, 42(1), 117–149. 17