Matrices That Commute with Their Conjugate and Transpose

ABSTRACT

It is known that if A∈M_{n} is normal (AA^{*}=A^{*}A) , then AA￣=A￣A if and only if AA^{T}=A^{T}A. This leads to the question: do both AA￣=A￣A and AA^{T}=A^{T}A imply that A is normal? We give an
example to show that this is false when n=4, but we show that it is true when n=2 and n=3.

Cite this paper

G. Goodson, "Matrices That Commute with Their Conjugate and Transpose,"*Advances in Linear Algebra & Matrix Theory*, Vol. 3 No. 3, 2013, pp. 22-25. doi: 10.4236/alamt.2013.33005.

G. Goodson, "Matrices That Commute with Their Conjugate and Transpose,"

References

[1] Kh. Ikramov, “On the Matrix Equation ,” Moscow University Computational Mathematics and Cybernetics. Vol. 34, No. 2, 2010, pp. 51-55. doi:10.3103/S0278641910020019

[2] G. R. Goodson and R. A. Horn, “Canonical Forms for Normal Matrices That Commute with Their Complex Conjugate,” Linear Algebra and Its Applications, Vol. 430, No. 4, 2009, pp. 1025-1038. doi:10.1016/j.laa.2008.09.039

[3] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, New York, 1985. doi:10.1017/CBO9780511810817

[1] Kh. Ikramov, “On the Matrix Equation ,” Moscow University Computational Mathematics and Cybernetics. Vol. 34, No. 2, 2010, pp. 51-55. doi:10.3103/S0278641910020019

[2] G. R. Goodson and R. A. Horn, “Canonical Forms for Normal Matrices That Commute with Their Complex Conjugate,” Linear Algebra and Its Applications, Vol. 430, No. 4, 2009, pp. 1025-1038. doi:10.1016/j.laa.2008.09.039

[3] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, New York, 1985. doi:10.1017/CBO9780511810817