Diagonal Representation of Certain Matrices
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1 Diagonal Representation of Certain Matrices Mark Tygert Research Report YALEU/DCS/RR-33 December 2, 2004 Abstract An explicit expression is provided for the characteristic polynomial of a matrix M of the form 0 ab T M = D ba T, () 0 where D is a diagonal matrix, and a and b are column vectors Also, an explicit expression is provided for the matrix of normalized eigenvectors of M, in terms of the roots of the characteristic polynomial (ie, in terms of the eigenvalues of M) A Lemma, a Remark, and an Observation The following lemma is verified by substituting into the left hand side of (7) the definitions of P in (6) and U in (9) (6), and simplifying the result using (4) See [2] for similar results, and [3] and [] for applications Lemma Suppose that m and n are positive integers, a = (a 0, a,, a m 2, a m ) T and b = (b 0, b,, b n 2, b n ) T are real vectors, and d 0, d,, d m+n 2, d m+n and λ 0, λ,, λ m+n 2, λ m+n are real numbers such that for any j, k (j, k = 0,,, m + n 2, m + n ), when j k, and ( m (a k ) 2 d k λ j (with j = 0,,, m + n 2, m + n ) λ j d k (2) λ j λ k (3) ) ( n (b k ) 2 Partially supported by the US DoD under a 200 NDSEG Fellowship ) = (4)
2 Report Documentation Page Form Approved OMB No Public reporting burden for the collection of information is estimated to average hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 25 Jefferson Davis Highway, Suite 204, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number REPORT DATE 2 DEC REPORT TYPE 3 DATES COVERED to TITLE AND SUBTITLE Diagonal Representation of Certain Matrices 5a CONTRACT NUMBER 5b GRANT NUMBER 5c PROGRAM ELEMENT NUMBER 6 AUTHOR(S) 5d PROJECT NUMBER 5e TASK NUMBER 5f WORK UNIT NUMBER 7 PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Yale University,Department of Computer Science,PO Box ,New Haven,CT, PERFORMING ORGANIZATION REPORT NUMBER 9 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 0 SPONSOR/MONITOR S ACRONYM(S) 2 DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 3 SUPPLEMENTARY NOTES 4 ABSTRACT 5 SUBJECT TERMS SPONSOR/MONITOR S REPORT NUMBER(S) 6 SECURITY CLASSIFICATION OF: 7 LIMITATION OF ABSTRACT a REPORT b ABSTRACT c THIS PAGE 8 NUMBER OF PAGES 4 9a NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev 8-98) Prescribed by ANSI Std Z39-8
3 Suppose further that D is the diagonal (m + n) (m + n) matrix defined by the formula D = d d, (5) dm+n d m+n and P is the (m + n) (m + n) matrix defined by the formula 0 ab T P = ba T, (6) 0 where 0 denotes matrices consisting entirely of zeroes Then, (D P ) U = U Λ, (7) where Λ is the diagonal (m + n) (m + n) matrix defined by the formula Λ = λ λ, (8) λm+n λ m+n and U is the orthogonal (m + n) (m + n) matrix defined by the formula AV R U = (9) BW S In (9), A is the diagonal m m matrix defined by the formula A = a a, (0) am a m B is the diagonal n n matrix defined by the formula b b B = bn b n, () 2
4 V is the m (m + n) matrix with entry V j,k defined by the formula V j,k = d j λ k (2) (with j = 0,,, m 2, m ; k = 0,,, m + n 2, m + n ), W is the n (m + n) matrix with entry W j,k defined by the formula W j,k = d m+j λ k (3) (with j = 0,,, n 2, n ; k = 0,,, m + n 2, m + n ), S is the diagonal (m+n) (m+n) matrix with the diagonal entries S 0,0, S,,, S m+n 2,m+n 2, S m+n,m+n defined by the formula / m 2 ( ) n ak c j 2 b k S j,j = +, (4) d k λ j and R is the diagonal (m + n) (m + n) matrix with the diagonal entries R 0,0, R,,, R m+n 2,m+n 2, R m+n,m+n defined by the formula R j,j = c j S j,j (5) In (4) and (5), c 0, c,, c m+n 2, c m+n are the real numbers defined by the formula c j = n (b k ) 2 (6) Remark 2 The equation (4) is equivalent to the characteristic (secular) equation det λ j I (D P ) = 0 (7) for the eigenvalues λ j (with j = 0,,, m + n 2, m + n ) of the matrix D P Observation 3 The upper block AV R of the matrix U defined in (9) has the form of a diagonal matrix (A) times a matrix of inverse differences (V ) times another diagonal matrix (R) The lower block BW S of the matrix U defined in (9) also has the form of a diagonal matrix (B) times a matrix of inverse differences (W ) times another diagonal matrix (S) Therefore, there exists an algorithm which applies such an N N matrix U (or its adjoint) to an arbitrary real vector of length N in O (N log(/ε)) operations, where ε is the precision of computations (see [3]) 2 Acknowledgements The author would like to thank V Rokhlin for many useful discussions and proofreading 3
5 References [] S Chandrasekaran and M Gu, A divide-and-conquer algorithm for the eigendecomposition of symmetric block-diagonal plus semiseparable matrices, Numerische Mathematik, 96 (2004), pp [2] M Gu and S C Eisenstat, A stable and efficient algorithm for the rank- modification of the symmetric eigenproblem, SIAM Journal of Matrix Analysis and Applications, 5 (994), pp [3], A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem, SIAM Journal on Matrix Analysis and Applications, 6 (995), pp
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