New bounds for the minimum eigenvalue of the Fan product of two $M$-matrices

Guanghui Cheng

Czechoslovak Mathematical Journal (2014)

• Volume: 64, Issue: 1, page 63-68
• ISSN: 0011-4642

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Abstract

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In this paper, we mainly use the properties of the minimum eigenvalue of the Fan product of $M$-matrices and Cauchy-Schwarz inequality, and propose some new bounds for the minimum eigenvalue of the Fan product of two $M$-matrices. These results involve the maximum absolute value of off-diagonal entries of each row. Hence, the lower bounds for the minimum eigenvalue are easily calculated in the practical examples. In theory, a comparison is given in this paper. Finally, to illustrate our results, a simple example is also considered.

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Cheng, Guanghui. "New bounds for the minimum eigenvalue of the Fan product of two $M$-matrices." Czechoslovak Mathematical Journal 64.1 (2014): 63-68. <http://eudml.org/doc/262026>.

@article{Cheng2014,
abstract = {In this paper, we mainly use the properties of the minimum eigenvalue of the Fan product of $M$-matrices and Cauchy-Schwarz inequality, and propose some new bounds for the minimum eigenvalue of the Fan product of two $M$-matrices. These results involve the maximum absolute value of off-diagonal entries of each row. Hence, the lower bounds for the minimum eigenvalue are easily calculated in the practical examples. In theory, a comparison is given in this paper. Finally, to illustrate our results, a simple example is also considered.},
author = {Cheng, Guanghui},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fan product; minimum eigenvalue; $M$-matrix; Fan product; minimum eigenvalue; -matrix; Cauchy-Schwarz inequality; lower bounds},
language = {eng},
number = {1},
pages = {63-68},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {New bounds for the minimum eigenvalue of the Fan product of two $M$-matrices},
url = {http://eudml.org/doc/262026},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Cheng, Guanghui
TI - New bounds for the minimum eigenvalue of the Fan product of two $M$-matrices
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 63
EP - 68
AB - In this paper, we mainly use the properties of the minimum eigenvalue of the Fan product of $M$-matrices and Cauchy-Schwarz inequality, and propose some new bounds for the minimum eigenvalue of the Fan product of two $M$-matrices. These results involve the maximum absolute value of off-diagonal entries of each row. Hence, the lower bounds for the minimum eigenvalue are easily calculated in the practical examples. In theory, a comparison is given in this paper. Finally, to illustrate our results, a simple example is also considered.
LA - eng
KW - Fan product; minimum eigenvalue; $M$-matrix; Fan product; minimum eigenvalue; -matrix; Cauchy-Schwarz inequality; lower bounds
UR - http://eudml.org/doc/262026
ER -

References

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1. Berman, A., Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics 9 SIAM, Philadelphia (1994). (1994) Zbl0815.15016MR1298430
2. Fang, M. Z., 10.1016/j.laa.2007.03.024, Linear Alegebra. Appl. 425 (2007), 7-15. (2007) Zbl1128.15011MR2334489DOI10.1016/j.laa.2007.03.024
3. Horn, R. A., Johnson, C. R., Topics in Matrix Analysis, Cambridge University Press Cambridge (1991). (1991) Zbl0729.15001MR1091716
4. Liu, Q. B., Chen, G. L., On two inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 431 (2009), 974-984. (2009) Zbl1183.15017MR2535567

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