Study on Hermitian, Skew-Hermitian and Uunitary Matrices as a part of Normal Matrices

Vishwa Nath Jha


A normal matrix plays an important role in the theory of matrices. It includes Hermitian matrices and enjoy several of the same properties as Hermitian matrices. Indeed, while we proved that Hermitian matrices are unitarily diagonalizable, we did not establish any converse; normal matrices are also unitarily diagonalizable. In this present paper we have tried to establish the proper relation of normal matrices with others. 

Full Text:



T. S. Motzkin and O. Taussky, Matrices with property L, Trans. Amer.

Math. Soc. vol. 73(1952), pp. 108-114.

I. Schur, Uber die characteristischem wurzeln einer linearen

substitutionen mit einer, Math. Ann. vol.66 (1909), pp. 488-510.

N. A. Wiegmann, A note on pairs of normal matrices with property L,

Amer. Math. Soc. (1952), pp. 35-36.

R. F. Rinchart, Skew matrices as square roots, Mathe. Asso. of

America, vol.2, No.2, (1960), pp.157-161.

R. Weitzenbock, Uber die matrixgleichung Nederl. Akad.

Wetensch. Proc., vol. 35, (1932), pp.157-161.

P. Franklin, Algebraic matrix equations, J. Math. Phys., vol. 10, (1932),

pp. 135-143.

K. Soda, Einige satze uber matrizen, Japan J. Math, 13(1936), pp. 361-

A. A. Albert and B. Muckenhoupt, On matrices of trace zero, Michigan

Math. J. 4(1957), 1- 3, MR 18, pp. 786.

C. R. Johnson, A note on matrix solutions to A = XY – YX, Amer.

Mathe. Soc. Vol. 42, No. 2(1974), pp. 351-353.

F. Gaines, A note on matrices with zero trace, Amer. Math. Monthly,

(1968), MR 33#7356, pp. 630-631.

W. V. Parker, Sets of complex numbers associated with a matrix,

Duke Mathe. J., 15(1948), MR 10, 230, pp. 711-715.

S. Friedland, Matices with prescribed off diagonal elements, Israel J.

Math. 11(1972), pp. 184-189.

G. Finke, R. E. Burkard and F. Rendl, Quadratic assignment

problems, Ann. Discrete Math., 31(1987), pp. 61-82.

E. R. Bernes and A. J. Hoffman, Bounds for the spectrum of normal

matrices, Linear Algebra and its Appl., 79(1994), pp. 79-90.

O. Taussky, A generalization of a theorem of Liapunov, J. Soc. Indust.

Appl. Math., 9(1961), pp. 640-643.

G. P. Barker, Normal matrices and the Lyapunov equations, SIAM J.

Appl. Math., vol. 21, No.1(1974), pp. 1-4.

A. Ostrowski and H. Schneider, Some theorems on the inertia of

general matrices, J. Math. Anal. Appl., 4(1962), pp. 72-84.

O. Taussky, A remark on a theorem of Lyapunov, J, Math. Anal.

Appl., 2(1961), pp. 105- 107.

G. P. Barker, Common solutions to the Lyapunov equations, Linear

Algebra and its Appl., 16(1977), pp. 233-235.

Robert Grone, C. R. Johnson, E. M. Sa. and H. Wolkowicz (GJSW),

Linear Algebra and its Appl., 87(1987), pp. 213-225.

L. Elsner and Kh. D. Ikramov, Normal matrices: An update, Linear

Algebra and its Appl., 285(1998), pp. 291-303.

M. Sadkane, A note on normal matrices, J. of Computational and

Applied Mathematics, 136(2001), pp. 185-187.

J. F. Querio, On the eigen values of normal matrices, Dept. of Mathe.,

Universidade de Coimbra, 3000 Coimbra, Portugal, 2001.

J. A. Ball, I. Gohberg and L. Rodman, Interpolation of rational matrix

functions, OT45, Birkhauser-Verlag, Basel, Switzerland, 1990.

H. Kimara, Chain-Scattering approach to control, Birkhauser

Boston, Boston, 1997.

N. J. Higham, J. orthogonal matrices: properties and generation, SIAM

Rev., 45(2003), pp. 505-519.

Yik-Hoi Au-Yeung, Chi-Cwong Li and Leiba Rodman, H-unitary and

Lorentz matrices: A review, SIAM J. Matrix Anal. Appl., vol. 25,

No.4(2004), pp. 1140-1162.

Kh. D. Ikramov, Conjugate normal matrices and matrix equations in

A, and AT , Doklady Mathematics, vol. 75, No.1(2007), pp. 55-

Kh. D. Ikramov, On condiagonalizable matrices, Linear Algebra and

its Appl., 429(2007), pp. 456-465.

R. Bhatia, Perturbation bounds for matrix eigen values, Wiley Pub.,

New York, 1987.

Kh. D. Ikramov, The condition of the coneigen values of conjugate-

normal matrices, Doklady Mathematics, vol. 77, No. 3(2008), pp.


R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University

Press, Cambridge, 1990.

H. Faβbender, Kh. D. Ikramov, Conjugate normal matrices: A survey,

Linear Algebra and its Appl., 429(2008), pp. 1425-1441.

Kh. D. Ikramov, Zh. Vychisl Mat. Mat. Fiz. 34 (1994), pp. 473-479.

Kh. D. Ikramov, Mat. Zametki 57, (1995), pp. 670-680.

Kh. D. Ikramov and V. N. Chugunov, Zh. Vychisl. Mat. Mat. Fiz. 36,

(1996), pp. 3-10.

Kh. D. Ikramov, Fandam. Prinkl. Mat. 3, (1997), pp. 809-819.

Kh. D. Ikramov and V. N. Chugunov, Veston. Mosk. Univ. Ser. 15,

Vychisl. Mat. Kibern, No. 1, (2007), pp. 10-13.

G. Gu and L. Patton, SIAM J. Matrix analysis. Appl. 24, (2003), pp.


Kh. D. Ikramov and V. N. Chugunov, Zap. Nauchn Semin. POMI

, (2007), pp. 63-80.

Kh. D. Ikramov and V. N. Chugunov, Classification normal Hankel

matrices, Doklady Mathematics, vol. 79, No. 1, (2009), pp. 114-117.

Miclo Ferranti and Ref Vandebril, Computing eigen values of normal

matrices via complex symmetric matrices, J. of Computational

Mathe., vol. 259, A(2013), pp. 281-293.

G. H. Colub, C. F. Van Loan, Matrix computations, 3 rd Edition,

Johns Hopkins University Press, Baltimore, Maryland, USA, 1996.

D. S. Watkins, The matrix eigen value problem, GR and Krylov

subspace methods, SIAM, Philadelphia, Pensylvania, USA, 2007.

Ref Vandebril, A unitary similarity transforms of a normal matrix to

complex symmetric from, Appl. Mathe. Letters, vol. 24, 2 (2011), pp.


C. G. Khatri, Powers of matrices and idempotency, Linear Algebra

Appl., 33 (1980), pp. 57- 65.

J. GroB, Idempotency of the hermitian part of a complex matrix,

Linear Algebra Appl., 289 (1999), pp. 135-139.

D. Ilisevic and N. Thome, When is the hermitian / skew hermitian part

of a matrix a potent matrix, J. of Linear Algebra, Int. Linear Algebra

Soc., vol. 24 (2012), pp. 94-112.

L. Brand, On the product of singular symmetric analysis, Proc. Amer.

Math. Soc., 22 (1969), p. 377.

D. Z. Djokovic, A determinantal inequality for projects in a unitary

space, Proc. Amer. Math. Soc., 27 (1971), pp. 19-23.

M. Lin, Orthogonal sets of normal or conjugate normal matrices,

Linear Algebra and its Appl., 483 (2015), pp. 227-235.


  • There are currently no refbacks.

Abava  Absolutech Convergent 2020

ISSN: 2307-8162