Advances in Computational Mathematics

, Volume 39, Issue 3–4, pp 585–609 | Cite as

Constructing all self-adjoint matrices with prescribed spectrum and diagonal

  • Matthew FickusEmail author
  • Dustin G. Mixon
  • Miriam J. Poteet
  • Nate Strawn


The Schur–Horn Theorem states that there exists a self-adjoint matrix with a given spectrum and diagonal if and only if the spectrum majorizes the diagonal. Though the original proof of this result was nonconstructive, several constructive proofs have subsequently been found. Most of these constructive proofs rely on Givens rotations, and none have been shown to be able to produce every example of such a matrix. We introduce a new construction method that is able to do so. This method is based on recent advances in finite frame theory which show how to construct frames whose frame operator has a given prescribed spectrum and whose vectors have given prescribed lengths. This frame construction requires one to find a sequence of eigensteps, that is, a sequence of interlacing spectra that satisfy certain trace considerations. In this paper, we show how to explicitly construct every such sequence of eigensteps. Here, the key idea is to visualize eigenstep construction as iteratively building a staircase. This visualization leads to an algorithm, dubbed Top Kill, which produces a valid sequence of eigensteps whenever it is possible to do so. We then build on Top Kill to explicitly parametrize the set of all valid eigensteps. This yields an explicit method for constructing all self-adjoint matrices with a given spectrum and diagonal, and moreover all frames whose frame operator has a given spectrum and whose elements have given lengths.


Schur–Horn Interlacing Majorization Frames 

Mathematics Subject Classification (2010)



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  1. 1.
    Antezana, J., Massey, P., Ruiz, M., Stojanoff, D.: The Schur–Horn theorem for operators and frames with prescribed norms and frame operator. Illinois J. Math. 51, 537–560 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Batson, J., Spielman, D.A., Srivastava, N.: Twice-Ramanujan sparsifiers. In: STOC 09: Proc. 41st Annu. ACM Symp. Theory Comput., pp. 255–262 (2009)Google Scholar
  3. 3.
    Bendel, R.B., Mickey, M.R.: Population correlation matrices for sampling experiments. Commun. Stat. Simul. Comput. 7, 163–182 (1978)CrossRefGoogle Scholar
  4. 4.
    Bodmann, B.G., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258, 397–420 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cahill, J., Fickus, M., Mixon, D.G., Poteet, M.J., Strawn, N.: Constructing finite frames of a given spectrum and set of lengths. Appl. Comput. Harmon. Anal. (to appear). arXiv:1106.0921
  6. 6.
    Casazza, P.G., Fickus, M., Mixon, D.G.: Auto-tuning unit norm tight frames. Appl. Comput. Harmon. Anal. 32, 1‒15 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Casazza, P.G., Fickus, M., Mixon, D.G., Wang, Y., Zhou, Z.: Constructing tight fusion frames. Appl. Comput. Harmon. Anal. 30, 175–187 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casazza, P.G., Kovačević, J.: Equal-norm tight frames with erasures. Adv. Comput. Math. 18, 387–430 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casazza, P.G., Leon, M.: Existence and construction of finite tight frames. J. Comput. Appl. Math. 4, 277–289 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chan, N.N., Li, K.-H.: Diagonal elements and eigenvalues of a real symmetric matrix. J. Math. Anal. Appl. 91, 562–566 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chu, M.T.: Constructing a Hermitian matrix from its diagonal entries and eigenvalues. SIAM J. Matrix Anal. Appl. 16, 207–217 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davies, P.I., Higham, N.J.: Numerically stable generation of correlation matrices and their factors. BIT 40, 640–651 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dhillon, I.S., Heath, R.W., Sustik, M.A., Tropp, J.A.: Generalized finite algorithms for constructing Hermitian matrices with prescribed diagonal and spectrum. SIAM J. Matrix Anal. Appl. 27, 61–71 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dykema, K., Strawn, N.: Manifold structure of spaces of spherical tight frames. Int. J. Pure Appl. Math. 28, 217–256 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Fickus, M., Mixon, D.G., Poteet, M.J.: In: Proc. SPIE 8138, 81380Q/1–8 (2011)Google Scholar
  16. 16.
    Goyal, V.K., Vetterli, M., Thao, N.T.: Quantized overcomplete expansions in \({\mathbb R}^N\): analysis, synthesis, and algorithms. IEEE Trans. Inform. Theory 44, 16–31 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Goyal, V.K., Kovačević, J., Kelner, J.A.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10, 203–233 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Higham, N.J.: Matrix nearness problems and applications. In: Gover, M.J.C., Barnett, S. (eds.) Applications of Matrix Theory, pp. 1–27. Oxford University Press (1989)Google Scholar
  19. 19.
    Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math. 76, 620–630 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  22. 22.
    Leite, R.S., Richa, T.R.W., Tomei, C.: Geometric proofs of some theorems of Schur–Horn type. Linear Algebra Appl. 286, 149–173 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Massey, P., Ruiz, M.: Tight frame completions with prescribed norms. Sampl. Theory Signal Image Process. 7, 1–13 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Schur, I.: Über eine klasse von mittelbildungen mit anwendungen auf die determinantentheorie. Sitzungsber. Berl. Math. Ges. 22, 9–20 (1923)Google Scholar
  25. 25.
    Strawn, N.: Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. J. Fourier Anal. Appl. 17, 821–853 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tropp, J.A., Dhillon, I.S., Heath, R.W.: Finite-step algorithms for constructing optimal CDMA signature sequences. IEEE Trans. Inform. Theory 50, 2916–2921 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Viswanath, P., Anantharam, V.: Optimal sequences and sum capacity of synchronous CDMA systems. IEEE Trans. Inform. Theory 45, 1984–1991 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York (outside the USA) 2013

Authors and Affiliations

  • Matthew Fickus
    • 1
    Email author
  • Dustin G. Mixon
    • 1
  • Miriam J. Poteet
    • 1
  • Nate Strawn
    • 2
  1. 1.Department of Mathematics and StatisticsAir Force Institute of TechnologyWright-Patterson Air Force BaseUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA

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