Appendix A. Rayleigh Ratios and the Courant-Fischer Theorem

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1 Appendix A Rayleigh Ratios and the Courant-Fischer Theorem The most important property of symmetric matrices is that they have real eigenvalues and that they can be diagonalized with respect to an orthogonal matrix. Thus, if A is an n n symmetric matrix, then it has n real eigenvalues 1,..., n (not necessarily distinct), and there is an orthonormal basis of eigenvectors (u 1,...,u n ) (for a proof, see Gallier [6]). 211

2 212 APPENDIX A. RAYLEIGH RATIOS AND THE COURANT-FISCHER THEOREM Another fact that is used frequently in optimization problem is that the eigenvalues of a symmetric matrix are characterized in terms of what is known as the Rayleigh ratio, definedby R(A)(x) = x> Ax x > x, x 2 Rn,x6= 0. The following proposition is often used to prove the correctness of various optimization or approximation problems (for example PCA).

3 Proposition A.1. (Rayleigh Ritz) IfA is a symmetric n n matrix with eigenvalues 1 apple 2 apple apple n and if (u 1,...,u n ) is any orthonormal basis of eigenvectors of A, where u i is a unit eigenvector associated with i, then 213 max x6=0 x > x = n (with the maximum attained for x = u n ), and max x6=0,x2{u n k+1,...,u n }? x > x = n k (with the maximum attained for x = u n 1 apple k apple n 1. k ), where Equivalently, if V k is the subspace spanned by (u 1,...,u k ), then k = max x6=0,x2v k x > x, k =1,...,n.

4 214 APPENDIX A. RAYLEIGH RATIOS AND THE COURANT-FISCHER THEOREM For our purposes, we also need the version of Proposition A.1 applying to min instead of max. Proposition A.2. (Rayleigh Ritz) IfA is a symmetric n n matrix with eigenvalues 1 apple 2 apple apple n and if (u 1,...,u n ) is any orthonormal basis of eigenvectors of A, where u i is a unit eigenvector associated with i, then min x6=0 x > x = 1 (with the minimum attained for x = u 1 ), and min x6=0,x2{u 1,...,u i 1 }? x > x = i (with the minimum attained for x = u i ), where 2 apple i apple n. Equivalently, if W k = V k? 1 denotes the subspace spanned by (u k,...,u n ) (with V 0 =(0)), then k = min x6=0,x2w k x > x = min x6=0,x2v? k 1 x > x, k =1,...,n.

5 215 Propositions A.1 and A.2 together are known as the Rayleigh Ritz theorem. As an application of Propositions A.1 and A.2, we give aproofofapropositionwhichisthekeytotheproofof Theorem 2.2. Given an n n symmetric matrix A and an m m symmetric B, withm apple n, if 1 apple 2 apple apple n are the eigenvalues of A and µ 1 apple µ 2 apple appleµ m are the eigenvalues of B, thenwesaythattheeigenvalues of B interlace the eigenvalues of A if i apple µ i apple n m+i, i =1,...,m. The following proposition is known as the Poincaré separation theorem.

6 216 APPENDIX A. RAYLEIGH RATIOS AND THE COURANT-FISCHER THEOREM Proposition A.3. Let A be an n n symmetric matrix, R be an n m matrix such that R > R = I (with m apple n), and let B = R > AR (an m m matrix). The following properties hold: (a) The eigenvalues of B interlace the eigenvalues of A. (b) If 1 apple 2 apple apple n are the eigenvalues of A and µ 1 apple µ 2 apple appleµ m are the eigenvalues of B, and if i = µ i, then there is an eigenvector v of B with eigenvalue µ i such that Rv is an eigenvector of A with eigenvalue i. Observe that Proposition A.3 implies that m apple tr(r > AR) apple n m n. The left inequality is used to prove Theorem 2.2.

7 For the sake of completeness, we also prove the Courant Fischer characterization of the eigenvalues of a symmetric matrix. 217 Theorem A.4. (Courant Fischer) Let A be a symmetric n n matrix with eigenvalues 1 apple 2 apple apple n and let (u 1,...,u n ) be any orthonormal basis of eigenvectors of A, where u i is a unit eigenvector associated with i.ifv k denotes the set of subspaces of R n of dimension k, then k = max min W 2V n k+1 x2w,x6=0 k =min W 2V k max x2w,x6=0 x > x x > x.

8 218 APPENDIX A. RAYLEIGH RATIOS AND THE COURANT-FISCHER THEOREM

9 Bibliography [1] Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation clustering. Machine Learning, 56:89 113, [2] Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15: ,2003. [3] Fan R. K. Chung. Spectral Graph Theory, volume 92 of Regional Conference Series in Mathematics. AMS, first edition, [4] Eric D. Demaine and Nicole Immorlica. Correlation clustering with partial information. In S. Arora et al., editor, Working Notes of the 6th International Workshop on Approximation Algorithms for Combinatorial Problems, LNCS Vol. 2764, pages Springer, [5] Jean H. Gallier. Discrete Mathematics. Universitext. Springer Verlag, first edition,

10 220 BIBLIOGRAPHY [6] Jean H. Gallier. Geometric Methods and Applications, For Computer Science and Engineering. TAM, Vol. 38. Springer, second edition, [7] Chris Godsil and Gordon Royle. Algebraic Graph Theory. GTMNo.207.SpringerVerlag,firstedition, [8] H. Golub, Gene and F. Van Loan, Charles. Matrix Computations. TheJohnsHopkinsUniversityPress, third edition, [9] Frank Harary. On the notion of balance of a signed graph. Michigan Math. J., 2(2): ,1953. [10] Jao Ping Hou. Bounds for the least laplacian eigenvalue of a signed graph. Acta Mathematica Sinica, 21(4): , [11] Ravikrishna Kolluri, Jonathan R. Shewchuk, and James F. O Brien. Spectral surface reconstruction from noisy point clouds. In Symposium on Geometry Processing, pages11 21.ACMPress,July2004. [12] Jérôme Kunegis, Stephan Schmidt, Andreas Lommatzsch, Jürgen Lerner, Ernesto William De Luca, and Sahin Albayrak. Spectral analysis of signed graphs for clustering, prediction and visualization. In SDM 10, pages ,2010.

11 BIBLIOGRAPHY 221 [13] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. Transactions on Pattern Analysis and Machine Intelligence, 22(8): , [14] Daniel Spielman. Spectral graph theory. In Uwe Naumannn and Olaf Schenk, editors, Combinatorial Scientific Computing. CRCPress,2012. [15] von Luxburg Ulrike. A tutorial on spectral clustering. Statistics and Computing, 17(4): ,2007. [16] Stella X. Yu. Computational Models of Perceptual Organization. PhD thesis, Carnegie Mellon University, Pittsburgh, PA 15213, USA, Dissertation. [17] Stella X. Yu and Jianbo Shi. Multiclass spectral clustering. In 9th International Conference on Computer Vision, Nice, France, October IEEE, 2003.

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