Juan-Miguel Gracia

Telephone: 34-945013116             Fax: 34-945013014

The University of the Basque Country, Department of Applied Mathematics and Statistics, Faculty of Pharmacy, 7 Paseo de la Universidad, ES-01006 Vitoria-Gasteiz, Spain

My photo in 2014

More photos


Spanish

Research:

General topics: Theory of Matrices, Matrix Analysis, Linear Algebra.

Special topics: spectral perturbation; pseudospectra; singular values; matrix functions; derivatives of eigenvalues; nearest matrices with less generic Jordan canonical form.



 

Manuscripts from 1996 on

1996

Safety neighbourhoods for the invariants of the matrix similarity (1996).

     Abstract: We give safety neighbourhoods for the necessary conditions in the change of  the Jordan  canonical form of a matrix under small perturbations. We also obtain the minimum distance from a n × n complex matrix which has less than k nonconstant invariant factors (k=2,...,n ) to the set of matrices which have more or equal to k. When k=2, we get in particular the distance from a nonderogatory matrix to the set of derogatory matrices.
rootjord.dvi.
 
 

1997

1998

Safety neighbourhoods for the Kronecker canonical form (1998).

     Abstract: We give safety neighbourhoods for the necessary conditions in the change of the Kronecker canonical form of a matrix pencil under small perturbations.
root98_2.dvi
 
 

Conjectures on pseudospectra, Cassini ovals and Laplacians (in Spanish) 1998. armonica.dvi.
 

1999


Smooth Jordanization (1999).

     Abstract: Let A be a square complex matrix function of class Cp defined on an open interval (a,b) of the real line. Then for all t in (a,b), the matrix A(t) is similar to a matrix Jt in Jordan form; so there exists an invertible matrix Pt such that Pt-1A(t)Pt=Jt. We give a partial answer to the question: Can the function that associates Pt with t be chosen of class Cp?
smojorda.ps.
 
 

Nearest pair with more nonconstant invariant factors. Pseudospectrum (1999).

     Abstract: Let (A,B) be a pair of complex matrices n × n y n × m. Suppose that the number of nonconstant (¹ 1) invariant factors of the polynomial matrix l[In,0]-[A,B] is less than k . For all complex number l denote by sn-(k-1) (l[In,0]-[A,B] ) the greatest (n-(k-1))th singular value of the matrix l[In,0]-[A,B].   The minimum absolute value of the real function of a complex variable l ® sn-(k-1) (l[In,0]-[A,B] ) gives the distance from (A,B) to the set of pairs with more or equal number of nonconstant invariant factors. When k=1, this specializes in the formula of Eising for the distance from a controllable pair (A,B) to the nearest uncontrollable pair.
    The complex numbers l lying in the subset level { l Î C : sn (l[In,0]-[A,B] ) £ e } are the uncontrollable modes of all the pairs that are within an e tolerance of (A,B).
uno99.dvi
 
 

Flows of matrix pencils (April 21, 1999).

     Abstract: We consider some matrix ordinary differential equations whose solutions preserve some features of the initial conditions; for example the Kronecker canonical form of a matrix pencil, or the Brunovsky canonical form of a matrix pair.
 flows.dvi
 
 

2000

2001

Geometric multiplicity margin for a submatrix  (February 19, 2001).

     Abstract: Let G be a square complex matrix with less than k nonconstant invariant factors. We find a complex matrix that gives an optimal approximation to G among all possible matrices that have more than or equal to k invariant factors, obtained by varying only the entries of a bottom right submatrix of G.
     Paper: margin.dvi, margin.ps, margin.pdf
     Screen: marginweb.pdf.
 
 




The multiplicative integral (in Spanish) October 21, 2001.

     Abstract: It is analyzed the relation of the multiplicative integral of Volterra with the logarithmic derivative and the linear differential systems.
     Screen: InteMult.pdf.
 
 



 
 


2002

xyz  (in Spanish) January 21, 2002 .

     Abstract: The key importance of the symbols for the comprehension of the mathematics is emphasized that, without them, they would be unintelligible. The evolution is analyzed of some of these signs. Likewise, the great utility is commented of the notation of Leibniz for the teaching of the infinitesimal calculus.
      Screen:  xyz.pdf.
      Paper: xyzp.pdf.
      Source: xyz.tex.
 
 

Linear algebra behind the search engines of Internet (in Spanish) November 11, 2002.

     Abstract: There are analyzed two applications of linear algebra to the search engines of Internet: the value assigned to every web page by Google (PageRank), and the latent semantic analysis.
     Screen: busca.pdf.
     Paper: buscap.pdf.

     If you have trouble to see the links that appear in busca.pdf, download the file defi.pdf, and put both files in the same folder.
 
 

Nonnegative matrices, random walks and Markov chains (in Spanish) November 11, 2002.

     Abstract: We give some definitions of digraph, Markov chain, random walk, as well as its relation with the theory of Perron Frobenius of nonnegative matrices. It contains programs in Matlab.
     Screen: defi.pdf.
     Paper: defip.pdf.
 
 

2003

2004

Multiplicities of pseudoeigenvalues, Presentation ILAS04, Coimbra, July 20, 2004.
     Abstract: Let A be an n-by-n complex matrix. For each ε > 0 let T be a connected component of the strict ε-pseudospectrum of A. The sum of the algebraic multiplicities of the eigenvalues of all matrices X such that || X - A || < ε that are inside T, is equal to the sum of the algebraic multiplicities of all eigenvalues that are inside T. Here ||·|| is the spectral norm.
    Screen: multpseudo.pdf.
     Alternative files for screen: mltpsd.pdf; + figures.pdf; or pvisible.pdf
   

Concept of pseudospectra

The pseudospectrum of an n-by-n complex matrix A, of level ε > 0 , is defined as the union of all spectra of the matrices X that are in the closed ball centered at A and radius ε in the matrix space, where the distance in this space is the one associated to the spectral norm. It is noted by Λε (A). Next you can see an animated gif image that show the evolution of Λε (A) when ε increases from 0.





This animation belongs to the matrix of order 3



whose eigenvalues are



The condition numbers of these (simple) eigenvalues are, respectively,



The three connected components of the level ε pseudospectrum grow so much more quickly the more big is the condition number of the eigenvalue of A that contain. In fact, if δj(ε) is the diameter of the connected component that contains the j-th eigenvalue, then its right derivative at 0 satisfies δ'j(0+)= 2 c(λj).




More information about pseudospectra of a matrix can be seen in




2005

Nearest matrix with two prescribed eigenvalues, link to the journal Linear Algebra and Its Applications.

2006

Stability of controlled invariant subspaces, with Francisco E. Velasco: link to the journal Linear Algebra and Its Applications.

2008

Nearest southeast submatrix that makes multiple a prescribed eigenvalue. Part 1, jointly with Francisco E. Velasco: link to the journal Linear Algebra and Its Applications.

2009

Multiplicities of the structured pseudoeigenvalues. arXiv:0907.1980v2 [math.SP]

Limits of the singular values of a pencil of matrices, (in Spanish) Limites.pdf (talk).





2010

Nearest southeast submatrix that makes multiple an eigenvalue of the normal northwest submatrix, pdf , jointly with Francisco E. Velasco

Lipschitz stability of controlled invariant subspaces, pdf , jointly with Francisco E. Velasco

Identical pseudospectra of any geometric multiplicity, pdf, jointly with Gorka Armentia and Francisco E. Velasco

Derivatives of the diameter and the area of a connected component of the pseudospectra, pdf, jointly with Gorka Armentia and Francisco E. Velasco

Second order pseudospectra of normal matrices, pdf, jointly with Gorka Armentia and Francisco E. Velasco




2014

My road to the spectral perturbation of matrices, pdf. Beamer presentation that contains my goodbye talk in the congress ALAMA-GAMM/ANLA'2014, July 14-16, Barcelona, Spain, dedicated to homage to me, Juan-Miguel Gracia, with ocassion of my retirement in 2015.

Also you can be here an Spanish excerpt of what I said in that conference.



 

Books that every student of Mathematics should read

  • Z.A. Melzak: Companion to Concrete Mathematics, Edit. Wiley, New York, 1973.
  • Z.A. Melzak: Mathematical Ideas, Modeling and Applications, Edit. Wiley, New York, 1976.



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Last update: August 27, 2016.