1 2007 no 49 2443 2449 Some Bounds for the Singular Values of Matrices Ramazan Turkmen and Haci Civciv Department of Mathematics Faculty of Art and Science Selcuk University 42031 Konya Turkey rturkmenselcukedutr hacicivcivselcukedut ID: 25279 Download Pdf

1 2007 no 49 2443 2449 Some Bounds for the Singular Values of Matrices Ramazan Turkmen and Haci Civciv Department of Mathematics Faculty of Art and Science Selcuk University 42031 Konya Turkey rturkmenselcukedutr hacicivcivselcukedut

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Applied Mathematical Sciences, Vol. 1, 2007, no. 49, 2443 - 2449 Some Bounds for the Singular Values of Matrices Ramazan Turkmen and Haci Civciv Department of Mathematics, Faculty of Art and Science Selcuk University, 42031 Konya, Turkey rturkmen@selcuk.edu.tr, hacicivciv@selcuk.edu.tr Abstract We know that to estimate matrix singular values ( especially the largest and the smallest ones ) is an attractive topic in matrix theory and numerical analysis. In this note, we ﬁrst provide a simple estimate for the smallest singular value )of positive deﬁnite matrix .

Secondly, we obtain some simple estimates for the smallest singular value ) and the largest singular value )ofany complex matrix , which is not necessarily positive deﬁnite. Finally, we get a simple estimate for the largest singular value )ofan nonsin- gular complex matrix . These estimates are presented as a function of the determinant and the Euclidean norm of and Mathematics Subject Classsiﬁcation: 15A18, 15A60, 15A15 Keywords: Singular values, matrix norm, determinant 1 Introduction Let be -by- matrix with complex (real) elements. We denote the smallest singular value of by

and its largest singular value by ). Using matrix norms, a simple upper bound of ) was given in [2]: Yu Yi-Sheng and Gu Dun-he [4], and G. Piazza and T. Politi [5] gave a simple lower bound of ) showing that if 2) is a nonsingular matrix, then 1) det

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2444 R. Turkmen and H. Civciv and det 2) respectively. In this paper, we ﬁrst provide a simple estimate for the smallest singular value )of positive deﬁnite matrix . We then obtain some simple estimates for the smallest singular value ) and the largest singular value )ofany complex matrix , which is not necessarily

positive deﬁnite. Finally, we get a simple estimate for the largest singular value )ofan nonsingular complex matrix 2 Preliminaries In this section, we review the basic results on matrices needed in this paper. For more comprehensive treatments on matrices we refer to [2]. Let be any matrix. The Eucledean norm of the matrix are deﬁned as i,j =1 ij (1) Also, the spectral norm of the matrix is max where is eigenvalue of and is conjugate transpose of the matrix If , , ..., are the eigenvalues of the matrix , then det ... (2) The sequare roots of the eigenvalues of are the singular

values of A. Since is Hermitian and positive semideﬁnite, the singular values of are real and nonnegative. This let us write them in sorted order ... If , , ..., are the singular values of the matrix , then =1 (3) Throughout this note, we denote the smallest singular value of by ), and its largest singular value by

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Bounds for singular values of matrices 2445 The arithmetic-geometric-mean inequality, or brieﬂy the AGM inequality is the most important inequality in the classical analysis. It simply states that if ,x , ..., x are nonnegative real numbers and , ,

..., 0 with =1 = 1, then =1 =1 and equality holds if and only if ... =1 The important unweighted case occurs if we put ... , ...x , ... (4) 3 Main Results Theorem 1 Let be any positive deﬁnite matrix. Then, Proof. From the arithmetic-geometric-mean inequality, we can write =1 =1 /n (5) and =1 =1 /n (6) Threfore, the inequalities (5) and (6) give =1 =1 (7) Thus, if we consider the identitiy =1 ) and the Ineq. (7), then we get =1

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2446 R. Turkmen and H. Civciv Consequently, we have an upper bound for the smallest singular value of the matrix such that This completes the

proof. Theorem 2 Let be any -by- complex matrix. Then, the smallest singular value and the largest singular value of satisfy 1) and +2 2 det ( )+2 Proof. The identity )+ ... ) give )+ ... =1 )+2 k>j (8) Thus, from this equality we obtain the inequality =1 )+ k>j =1 =1 =1 =1 =1 =1 (9) Hence, the Ineq. (9) implies that =1 k< (10)

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Bounds for singular values of matrices 2447 By solving the Ineq. (10) for ), we get 1) To obtain a lower bound for the largest singular value of , let us consider the equality (8). Therefore, we write =1 k>j (11) If we use in (11) the inequality =1 1+

1+ =1 )+ i then we obtain +2 =1 =1 1+ +2 =1 (12) Note that ), =1 , ..., n , are the eigenvalues of (with associated eigenvectors ). Then, for each Ix Ax 1+ Therefore, =1+ ), =1 , ..., n , are the eigenvalues of the matrix . Hence, we can write det ( )= =1 1+ (13) Combining (12) and (13), we obtain +2 2 det ( )+2 n (14) We solve the inequality (14) for ) to obtain +2 2 det ( )+2

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2448 R. Turkmen and H. Civciv Theorem 3 Let be an 3) nonsingular complex matrix. Then, the largest singular value of satisﬁes (det (2 Proof. Using the artihmetic-geometric-mean inequality, we can

easily write )+ ... )+ )+ ... (15) On the other hand, to obtain an upper bound for the largest singular value of , we now will apply the artihmetic-geometric-mean inequality on the product ) det . Hence, we have ) (det ... ... )+ ... )+ ... )) (16) From (15) and (16), we get ) (det (17) Consequently, from (17) we ﬁnd an upper bound for the largest singular value of such that (det (2 References [1] C. R. Johnson and T. Szulc, Further lower bounds for the smallest singular value , Linear Alg. and Its Appl., 272, 169-179, 1998.

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Bounds for singular values of matrices 2449

[2] R. A. Horn, and C. R. Johnson, Topics in matrix analysis , Cambridge University Press, New York, 1991, [3] O. Rojo, R. Soto and H. Rojo, Bounds for the spectral radius and the largest singular value, Computers Math. Appl., 36, 1, 41-50, 1998. [4] Yu Yi-Sheng and Gu Dun-he, A note on a lower bound for the smallest singular value , Linear Alg. and Its Appl., 253, 25-38, 1997. [5] G. Piazza and T. Politi, An upper bound for the condition number of a matrix in spectral norm , Journal of Computational and Appl. Math., 143, 141-144, 2002. Received: April 17, 2007

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