Numerical range
  1. M. T. Chien, Boundedness of the numerical range, Linear Algebra and Its Applications, 134(1990), 25-30.

  2. M. T. Chien and B. S. Tam, Circularity of the numerical range, Linear Algebra and Its Applications 201(1994), 113-133.

  3. M. T. Chien, Numerical computation and circularity of higher numerical range, Soochow Journal Mathematics 20(1994), 555-567.

  4. M. T. Chien, On the numerical range of tridiagonal operators, Linear Algebra and Its Applications 246(1996), 203-214.

  5. M. T. Chien, The envelope of the generalized numerical range, Linear and Multilinear Algebra 43(1998), 363-376.

  6. M. T. Chien, Lina Yeh, Y. T. Yeh and F. Z. Lin, On geometric properties of the numerical range, Linear Algebra and Its Applications 274(1998), 389-410.

  7. M. T. Chien and M. Neumann, Positive definiteness of tridiagonal matrices via the numerical range, Electronic Journal of Linear Algebra 3(1998), 93-102.

  8. M. T. Chien and H. Nakazato, Boundary generating curves of the c-numerical range, Linear Algebra and Its Applications, 294(1999), 67-84.

  9. M. T. Chien and Y. H. Lin, On the area of numerical range, Soocchow Journal of Mathematics, 26(2000), 255-269.

  10. M. T. Chien and J. M. Huang, Numerical range of a continuant matrix, Applied Mathematics Letters, 14(2001), 213-216.

  11. M. T. Chien, The c-numerical range of a rank one matrix, Applied Mathematics Letters, 14(2001), 167-170.

  12. M. T. Chien and H. Nakazato, The c-numerical range of tridiagonal matrices, Linear Algebra and Its Applications, 335(2001), 55-61.

  13. M. T. Chien and H. Nakazato, Davis-Wielandt shell and q-numerical range, Linear Algebra and Its Applications, 340(2002), 15-31.

  14. M. T. Chien and H. Nakazato, The numerical range of linear pencils of 2-by-2 matrices, Linear Algebra and Its Applications, 341(2002), 69-100.

  15. M. T. Chien, H. Nakazato and P. Psarrakos, Point equation of the boundary of the numerical range of a matrix polynomial, Linear Algebra and Its Applications, 347(2002), 205-217.

  16. Y. A. Alpin, M. T. Chien and L. Yeh, The numerical radius and bounds for zeros of a polynomial, Proceedings American Mathematical Society, 131(2003), 725-730.

  17. M. T. Chien, S. H. Tso and P. Y. Wu, Higher-dimensional numerical ranges of quadratic operators, Journal of Operator Theory, 49(2003), 155-173.

  18. M. T.  Chien and H. Nakazato, Complex formulation of Poncelet property in numerical range, Linear and Multilinear Algebra, 52(2004), 159-175.

  19. M. T.  Chien, H. Nakazato and P. Psarrakos, On the q-numerical range of matrices and matrix polynomials,  Linear and Multilinear Algebra, 53(2005), 357-374.

  20. M. T. Chien and H. Nakazato, The q-numerical ranges of normal operators, Linear and Multilinear Algebra, 53(2005), 393-416.

  21. M. T. Chien, H. Nakazato and P. Psarrakos, The q-numerical range and the Davis-Wielandt shell of reducible 3-by-3 matrices, Linear and Multilinear Algebra, 54(2006), 79-112.

  22. M. T. Chien and H. Nakazato, Elliptic Curves Arising from Numerical Ranges, International Journal of Pure and Applied Mathematics, 30(2006), 441-465.

  23. M. T. Chien and H. Nakazato, The q-numerical range of a reducible matrix via a normal operator, Linear Algebra and Its Applications, 419(2006), 440-465.

  24. M. T. Chien and H. Nakazato, The q-numerical radius of weighted shift operators with periodic weights, Linear Algebra and Its Applications, 422(2007), 198-218.

  25. M. T. Chien and H. Nakazato, The boundary of the q-numerical range of a reducible matrix, Linear and Multilinear Algebra, 55(2007), 275-292.

  26. M. T. Chien and H. Nakazato, Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices, Linear and Multilinear Algebra, 56(2008), 143-162.

  27. M. T. Chien and H. Nakazato, The q-numerical range of unitarily irreducible 3-by-3 matrices, International Journal of Contemporary Mathematical Sciences, 3(2008), 339-355.

  28. M. T. Chien and H. Nakazato, Perturbation of the q-numerical radius of a weighted shift operator, Journal of Mathematical Analysis and Applications, 345(2008), 954-963.

  29. M. T. Chien and H. Nakazato, The q-numerical range of 3-by-3 weighted shift matrices, Applied Mathematics Letter, 21( 2008), 1199-1203.

  30. M. T. Chien and H. Nakazato, Flat portions on the boundary of the Davis-Wielandt shell of 3-by-3 matrices, Linear Algebra and Its Applications, 430(2009), 204-214.

  31. M. T. Chien and H. Nakazato, The numerical radius of a weighted shift operator with geometric weights, Electronic Journal of Linear Algebra, 18(2009), 58-63.

  32. M. T. Chien and H. Nakazato, Joint numerical range and its generating hypersurface, Linear Algebra and Its Applications, 432(Jan 2010), 173-179.

  33. M. T. Chien and L. Yeh, On the boundary of the numerical range of a matrix, Applied Mathematics Letters, 23(June 2010), 725-727.

  34. M. T. Chien, C. L. Ko and H. Nakazato, On the numerical ranges of matrix products, Applied Mathematics Letters, 23(June 2010), 732-737.

  35. M. T. Chien and H. Nakazato, The q-numerical range of 3×3 tridiagonal matrices, Electronic Journal of Linear Algebra, 20(July, 2010), 376-390.

  36. M. T. Chien and H. Nakazato, Numerical range for orbits under a central force, Mathematical Physics Analysis and Geometry, 13(Dec 2010), 315-330.

  37. M. T. Chien and H. Nakazato, The numerical range of a tridiagonal operator, Journal of Mathematical Analysis and Applications, 373 (January, 2011), 297-304.

  38. M. T. Chien and H. Nakazato, Reduction of the c-numerical range to the classical numerical range, Linear Algebra and Its Applications, 434(Feb 2011), 615-624.

  39. M. T. Chien, Lucas' theorem and numerical range, Linear and Multilinear Algebra, 59( June 2011), 687-691.

  40. M. T. Chien and H. Nakazato, Construction of determinantal representation of trigonometric polynomials, Linear Algebra and Its Applications, 435(Sept 2011), 1277-1284.

  41. M. T. Chien and H. Nakazato, The boundary of higher rank numerical ranges, Linear Algebra and Its Applications, 435( Dec 2011), 2971-2985.

  42. M. T. Chien and H. Nakazato, Central force and the higher rank numerical range, Journal of Mathematical Analysis and Applications, 389(May 2012), 531-540.

  43. M. T. Chien and K. H. Hung, Elliptic numerical ranges of bordered matrices, Taiwanese  Journal Mathematics, 16(June 2012), 1007-1016.

  44. M. T. Chien and H. Nakazato, Determinantal representations of closed orbits, Linear Algebra and Its Applications, 437(Aug 2012), 992-1002.

  45. M. T. Chien and H. Nakazato, Singular points of the ternary polynomials associated with 4 by 4 matrices, Electronic Journal of Linear Algebra, 23(Aug 2012), 755-769.

  46. M. T. Chien and H. Nakazato, Critical values for higher rank numerical ranges associated with roulette curves, Linear Algebra and Its Applications,  437(Nov 2012), 2117-2127.

  47. M. T. Chien and H. Nakazato, Strict convexity of the joint c-numerical range, Linear Algebra and Its Applications,  438(Feb 2013), 1305-1321.

  48. M. T. Chien and H. A. Sheu, The numerical radii of weighted shift matrices and operators, Operators and Matrices, 7(March 2013), 197-204.

  49. M. T. Chien and H. Nakazato, Cubic surfaces and q-numerical ranges, Mathematical Communications, 18(May 2013), 133-141.

  50. M. T. Chien and H. Nakazato, Hyperbolic forms associated with cyclic weighted shift matrices, Linear Algebra and Its Applications, 439(Dec 2013), 3541-3554.

  51. M. T. Chien and H. Nakazato, Singular points of cyclic weighted shift matrices, Linear Algebra and Its Applications, 439(Dec 2013), 4090-4100.

  52. M. T. Chien and H. Nakazato, Determinantal representation of trigonometric polynomial curves via Sylvester method, Banach Journal of Mathematical Analysis, 8(Jan 2014), 269-278.

  53. M. T. Chien and H. Nakazato, Reduction of joint c-numerical ranges, Applied Mathematics and Computation, 232(Apr 2014), 178-182.

  54. M. T. Chien and H. Nakazato, Numerical range of a nilpotent matrix and related central force,  Linear and Multilinear Algebra, 62(May 2014), 595-613.

  55. M. T. Chien and H. Nakazato, Singular points of the algebraic curves of symmetric hyperbolic forms, Linear Algebra and Its Applications, 470(April 2015), 40-50.

  56. M. T. Chien and H. Nakazato, Determinantal representations of hyperbolic forms via weighted shift matrices, Applied Mathematics and Computation, 258(May 2015), 172-181.

  57. M. T. Chien and H. Nakazato, Elliptic modular invariants and numerical ranges,  Linear and Multilinear Algebra, 63(Aug 2015), 1501-1519.

  58. B. Undrakh, H. Nakazato, A. Vandanjav, M. T. Chien*, The numerical radius of a weighted shift operator, Electronic Journal of Linear Algebra, 30(Dec 2015), 944-963.

  59. M. T. Chien and H. Nakazato, A new Poncelet curve for the boundary generating curve of a numerical range, Linear Algebra and Its Applications, 487(Dec 2015), 1-21.

  60. M. T. Chien, H. Nakazato, B. Undrakh and A. Vandanjav, Determinantal polynomials of a weighted shift operator, Linear and Multilinear Algebra, 64(Jan 2016), 2-13.

  61. M. T. Chien, H.-L. Gau*, C.-K. Li, M.-C. Tsai, K.-Z. Wang, Product of operators and numerical range, Linear and Multilinear Algebra, 64(Jan 2016), 58-67.

  62. M. T. Chien, S.-T. Fang, Y.-X. Su, A generalization of Gershgorin circles, Applied and Computational Mathematics, 15(Feb 2016), 106-111.

  63. M. T. Chien and H. Nakazato, Computing the determinantal representations of hyperbolic forms, Czechoslovak Mathematical Journal, 66(Sept 2016), 633-651.

  64. M. T. Chien and H. Nakazato, Singular points of the algebraic curves associated to unitary bordering matrices, Linear Algebra and Its Applications, 513(Jan 2017), 224-239.

  65. M. T. Chien and H. Nakazato, Computation of Riemann matrices for the hyperbolic curves of determinantal polynomials, Annals of Functional Analysis, 8(May 2017), 152-167.

  66. M. T. Chien and H. Nakazato, Reducibility of the ternary forms of unitary bordering matrices, Linear Algebra and Its Applications, 527(Aug 2017), 73-86.

  67. M. T. Chien, J, Liu, H. Nakazato and T.-Y. Tam, Toeplitz matrices are unitarily similar to symmetric matrices, Linear and Multilinear Algebra, 65(Oct 2017), 2131–2144.

  68. M. T. Chien and H. Nakazato, Unitary similarity of the determinantal representation of unitary bordering matrices, Linear Algebra and Its Applications, 541(March 2018), 13-35.

  69. M. T. Chien and H. Nakazato, Symmetric representation of ternary forms associated to some Toeplitz matrices, Symmetry, 10(March, 2018), 55; doi:10.3390/sym10030055.

  70. M. T. Chien and H. Nakazato, Determinantal representations of elliptic curves via Weierstrass elliptic functions, Electronic Journal of Linear Algebra, 34(March, 2018), 125-136.

c*-algebra

  1. M. T. Chien, C*-algebra fibre bundles, Transactions American Mathematical Society, 273(1982), 795-801.

  2. M. T. Chien, Duality of C*-algebra fibre bundles, Mathematical Reports, Academy Science Canada, 4(1982), 331-336.

  3. M. T. Chien, Tensor products of C*-algebra fibre bundles, Mathematical Reports, Academy Science Canada, 6(1984), 211-216.

  4. M. T. Chien, Algebra actions of semigroup algebras, Soochow Journal of Mathematics, 10(1984), 33-48.

  5. M. T. Chien, Closed derivations on the unit square, Bulletin Australian Mathematics Society, 11(1986), 55-67.

  Miscellanea

  1. M. T. Chien, Generalized convolutions, Chinese Journal of Mathematics, 10(1982), 31-38.

  2. M. T. Chien, On the adjoint of spectral operators, Bulletin Institute of Mathematics, Academia Sinica, 11(1983), 257-260

  3. M. T. Chien, The Fourier transforms of almost periodic functions on groups, Chinese Journal of Mathematics, 19(1991), 1-10.

  4. W. H Chen and M. T. Chien, Tree sign pattern matrices that require zero eigenvalues, Bulletin Australian Mathematical Society 55(1997), 81-88.

  5. M. T. Chien, Independent edges of certain graphs, Combinatorial Mathematics and Combinatorial Computing, 30(1999), 231-236.

  6. M. T. Chien and H. Nakazato, Circumscribed sphere of a convex polyhedron, Applied Mathematics Letters, 18(2005), 1199-1203.



 

 c-numerical range plotting

    The following codes plot the boundary of the c-numerical range of a matrix


       

    The image is a sample run for A=[1 0 1;0 -1 2;0 0 -1]; c=[1 0 0]; m=50

 clg;clc;
 disp('');disp('');disp('');disp('');
 disp('Programmer: Mao-Ting Chien');
disp(' Department of Mathematics');
disp(' Soochow University');
disp(' Taipei, TAIWAN');
disp(' e-mail: mtchien@scu.edu.tw');
disp(' ');
disp(' Save this file  as cnumg.m');
disp('1. run matlab');
disp('2. input this file name  cnumg');
disp(' ')
disp(' This program produces the boundary of the c-numerical range of a ');
disp('complex matrix by using MATLAB package, where c is a real n-tuple.');
disp('Enter a complex matrix A, an n-tuple c and the number of ');
disp('boundary points m');
disp('eg. A=[1 2;3 4] enter, c=[5 6] enter, m=50 enter ');
disp(' ');
disp(' ****** Press any key to run the program ******');
pause;
axis;
axis('square')
i=sqrt(-1);
Tol=10^(-5);
A=input('enter a matrix A= ');
C=input('enter an n-tuple c= ');
m=input('enter steps m= ');
% The begin of progran
C=-(sort(-C));
p=[];
for j=1:m+1
theta=(j-1)*2*pi/m;
H=(exp(i*theta)*A+exp(-i*theta)*A')/2;
[X,L]=eig(H);
[lambda,s]=sort(-diag(real(L)));
lambda=-lambda;
X=X(:,s);
k=1;
Y=[];
while k<size(A)
E=[];
if abs(lambda(k)-lambda(k+1)) > Tol
E=X(:,k)/norm(X(:,k));
Y=[Y E];
if k==size(A)-1
E=X(:,k+1)/norm(X(:,k+1));
Y=[Y E];
end
k=k+1;
else
E=[E X(:,k)];
while abs(lambda(k)-lambda(k+1)) <Tol
E=[E X(:,k+1)];
k=k+1;
if (k>=size(A)),break,end
end
E=orth(E);
Y=[Y E];
if k==size(A)-1
E=X(:,k+1)/norm(X(:,k+1));
Y=[Y E];
end
k=k+1;
end
end
for k=1:size(A)
S=Y(:,k)'*A*Y(:,k);
Q(k)=C(k)*S;
end
p(j)=sum(Q);
end
plot(real(p),imag(p));
meta cnumg
% end of the program