Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices

ABSTRACT

A new inequality on the minimum eigenvalue for the Fan product of nonsingular*M*-matrices is given. In addition, a new
inequality on the spectral radius of the Hadamard product of nonnegative
matrices is also obtained. These inequalities can improve considerably some
previous results.

A new inequality on the minimum eigenvalue for the Fan product of nonsingular

KEYWORDS

*M*-Matrix,
Nonnegative Matrix,
Fan Product,
Hadamard Product,
Spectral Radius,
Minimum Eigenvalue

Cite this paper

Gao, D. (2015) Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices.*Advances in Linear Algebra & Matrix Theory*, **5**, 90-97. doi: 10.4236/alamt.2015.53009.

Gao, D. (2015) Matrix Inequalities for the Fan Product and the Hadamard Product of Matrices.

References

[1] Berman, A. and Plemmons, R.J. (1979) Nonnegaive Matrices in the Mathematical Sciences. Academic Press, New York.

[2] Horn, R.A. and Johnson, C.R. (1985) Topics in Matrix Analysis. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511810817

[3] Fang, M.Z. (2007) Bounds on Eigenvalues for the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 425, 7-15.

http://dx.doi.org/10.1016/j.laa.2007.03.024

[4] Huang, R. (2008) Some Inequalities for the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 428, 1551-1559.

http://dx.doi.org/10.1016/j.laa.2007.10.001

[5] Li, Y.T., Li, Y.Y., Wang, R.W. and Wang, Y.Q. (2010) Some New Lower Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 432, 536-545.

http://dx.doi.org/10.1016/j.laa.2009.08.036

[6] Liu, Q.B. and Chen, G.L. (2009) On Two Inequalities for the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 431, 974-984.

http://dx.doi.org/10.1016/j.laa.2009.03.049

[7] Liu, Q.B., Chen, G.L. and Zhao, L.L. (2010) Some New Bounds on the Spectral Radius of Matrices. Linear Algebra and its Applications, 432, 936-948.

http://dx.doi.org/10.1016/j.laa.2009.10.006

[8] Zhou, D.M., Chen, G.L., Wu, G.X. and Zhang, X.Y. (2013) On Some New Bounds for Eigenvalues of the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 438, 1415-1426.

http://dx.doi.org/10.1016/j.laa.2012.09.013

[9] Zhao, L.L. (2012) Two Inequalities for the Hadamard Product of Matrices. Journal of Inequalities and Applications, 2012, 1-7.

http://dx.doi.org/10.1186/1029-242X-2012-122

[10] Varga, R.S. (1962) Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs.

[11] Berman, A. and Plemmons, R.J. (1994) Nonnegaive Matrices in the Mathematical Sciences. SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9781611971262

[12] Brauer, A. (1947) Limits for the Characteristic Roots of a Matrix Ⅱ. Duke Mathematical Journal, 14, 21-26.

http://dx.doi.org/10.1215/S0012-7094-47-01403-8

[1] Berman, A. and Plemmons, R.J. (1979) Nonnegaive Matrices in the Mathematical Sciences. Academic Press, New York.

[2] Horn, R.A. and Johnson, C.R. (1985) Topics in Matrix Analysis. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511810817

[3] Fang, M.Z. (2007) Bounds on Eigenvalues for the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 425, 7-15.

http://dx.doi.org/10.1016/j.laa.2007.03.024

[4] Huang, R. (2008) Some Inequalities for the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 428, 1551-1559.

http://dx.doi.org/10.1016/j.laa.2007.10.001

[5] Li, Y.T., Li, Y.Y., Wang, R.W. and Wang, Y.Q. (2010) Some New Lower Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 432, 536-545.

http://dx.doi.org/10.1016/j.laa.2009.08.036

[6] Liu, Q.B. and Chen, G.L. (2009) On Two Inequalities for the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 431, 974-984.

http://dx.doi.org/10.1016/j.laa.2009.03.049

[7] Liu, Q.B., Chen, G.L. and Zhao, L.L. (2010) Some New Bounds on the Spectral Radius of Matrices. Linear Algebra and its Applications, 432, 936-948.

http://dx.doi.org/10.1016/j.laa.2009.10.006

[8] Zhou, D.M., Chen, G.L., Wu, G.X. and Zhang, X.Y. (2013) On Some New Bounds for Eigenvalues of the Hadamard Product and the Fan Product of Matrices. Linear Algebra and its Applications, 438, 1415-1426.

http://dx.doi.org/10.1016/j.laa.2012.09.013

[9] Zhao, L.L. (2012) Two Inequalities for the Hadamard Product of Matrices. Journal of Inequalities and Applications, 2012, 1-7.

http://dx.doi.org/10.1186/1029-242X-2012-122

[10] Varga, R.S. (1962) Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs.

[11] Berman, A. and Plemmons, R.J. (1994) Nonnegaive Matrices in the Mathematical Sciences. SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9781611971262

[12] Brauer, A. (1947) Limits for the Characteristic Roots of a Matrix Ⅱ. Duke Mathematical Journal, 14, 21-26.

http://dx.doi.org/10.1215/S0012-7094-47-01403-8