Abstract
A square matrix A over the field of complex numbers is said to be Hermitian if \(A = A^{*}\), the conjugate transpose of A, while Hermitian matrices are known to be an important class of matrices. In addition to the definition, a Hermitian matrix can be characterized by some other matrix equalities. This fact can be described in the implication form \(f(A, A^{*}) = 0 \Leftrightarrow A = A^{*}\), where \(f(\cdot )\) denotes certain ordinary algebraic operation of A and \(A^{*}\). In this note, we show two special cases of the equivalent facts: \(AA^{*}A = A^{*}AA^{*} \Leftrightarrow A^3 = AA^{*}A \Leftrightarrow A = A^{*}\) without assuming the invertibility of A through the skillful use of decompositions and determinants of matrices. Several consequences and extensions are presented to a selection of matrix equalities composed of multiple products of A and \(A^{*}\).
Similar content being viewed by others
References
Baksalary, O.M., Trenkler, G.: Characterizations of EP, normal, Hermitian matrices. Linear Multilinear Algebra 5, 299–304 (2008)
Basavappa, P.: On the solutions of the matrix equation \(f(X,\, X^*)=g(X,\, X^*)\). Canad. Math. Bull. 15, 45–49 (1972)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)
Bernstein, D.S.: Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas-Revised and, Expanded Princeton University Press, Princeton (2018)
Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. SIAM, Philadelphia (2009)
Dehimi, S., Mortad, M.H., Tarcsay, Z.: On the operator equations \(A^n = A^{\ast }A\). Linear Multilinear Algebra 69, 1771–1778 (2021)
Djordjević, D.S., Koliha, J.J.: Characterizing Hermitian, normal and EP operators. Filomat 21, 39–54 (2007)
Erdelyi, I.: On the “reverse order law’’ related to the generalized inverse of matrix products. J. Assoc. Comp. Mach. 13, 439–443 (1966)
Erdelyi, I.: Partial isometries closed under multiplication on Hilbert spaces. J. Math. Anal. Appl. 22, 546–551 (1968)
Greville, T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518–521 (1966); and Erratum, SIAM Rev. 9(1967), 249
Hartwig, R.E., Spindelböck, K.: Matrices for which \(A^{\ast }\) and \(A^{\dagger }\) can commute. Linear Multilinear Algebra 14, 241–256 (1983)
Laberteux, K.R.: Problem 10377: Hermitian matrices. Amer. Math. Monthly 101, 362 (1993)
Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974)
McCullough, S.A., Rodman, L.: Hereditary classes of operators and matrices. Am. Math. Mon. 104, 415–430 (1997)
Mosić, D., Djordjević, D.S.: Moore-Penrose-invertible normal and Hermitian elements in rings. Linear Algebra Appl. 431, 732–745 (2009)
Mosić, D., Djordjević, D.S.: New characterizations of EP, generalized normal and generalized Hermitian elements in rings. Appl. Math. Comput. 218, 6702–6710 (2012)
Sebestyén, Z., Tarcsay, Zs.: Characterizations of selfadjoint operators. Studia Sci. Math. Hungar. 50, 423–435 (2013)
Sebestyén, Z., Tarcsay, Zs.: Characterizations of essentially selfadjoint and skew-adjoint operators. Studia Sci. Math. Hungar. 52, 371–385 (2015)
Smith, M.I.: A Schur algorithm for computing matrix \(p\)th roots. SIAM J. Matrix Anal. Appl. 24, 971–989 (2003)
Tian, Y.: A family of 512 reverse order laws for generalized inverses of a matrix product: a review. Heliyon 6, e04924 (2020)
Tian, Y.: Equivalence analysis of different reverse order laws for generalized inverses of a matrix product. Indian J. Pure Appl. Math. 53, 939–947 (2022)
Tian, Y.: A study of range equalities for matrix expressions that involve matrices and their generalized inverses. Comput. Appl. Math. 41, 384 (2022)
Wang, B., Zhang, F.: Words and normality of matrices. Linear Multilinear Algebra 40, 111–118 (1995)
Yan, S., Zhu, J.: On operator equation \(\lambda A^2 + \mu A^{*2} = \alpha A^{\ast }A + \beta AA^{\ast }\). Sci. Sinica Ser. A. 31, 531–539 (1998)
Zhang, F.: Matrix Theory: Basic Results and Techniques, 2nd edn. Springer, New York (2011)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that there are no competing interests.
Ethical Approval
Not applicable.
Additional information
Communicated by Daniel Alpay.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
“This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.”.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tian, Y. Some New Characterizations of a Hermitian Matrix and Their Applications. Complex Anal. Oper. Theory 18, 2 (2024). https://doi.org/10.1007/s11785-023-01440-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-023-01440-x