Applied Mathematical Sciences, Vol., 007, no. 49, 443-449 Some Bounds for the Singular Values of Matrices Ramazan Turkmen and Haci Civciv Department of Mathematics, Faculty of Art and Science Selcuk University, 403 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 σ n (A) ofn n positive definite matrix A. Secondly, we obtain some simple estimates for the smallest singular value σ n (A) and the largest singular value σ (A) ofanyn n complex matrix A, which is not necessarily positive definite. Finally, we get a simple estimate for the largest singular value σ (A) ofann n nonsingular complex matrix A. These estimates are presented as a function of the determinant and the Euclidean norm of A and n. Mathematics Subject Classsification: 5A8, 5A60, 5A5 Keywords: Singular values, matrix norm, determinant Introduction Let A be n-by-n matrix with complex (real) elements. We denote the smallest singular value of A by σ n (A), and its largest singular value by σ (A). Using matrix norms, a simple upper bound of σ (A) was given in []: σ (A) [ ] /. Yu Yi-Sheng and Gu Dun-he [4], and G. Piazza and T. Politi [5] gave a simple lower bound of σ n (A) showing that if A C n n (n ) is a nonsingular matrix, then ( ) (n )/ n σ n (A) det A, E
444 R. Turkmen and H. Civciv and σ n (A) det A (n )/ E, respectively. In this paper, we first provide a simple estimate for the smallest singular value σ n (A) ofn n positive definite matrix A. We then obtain some simple estimates for the smallest singular value σ n (A) and the largest singular value σ (A) ofanyn n complex matrix A, which is not necessarily positive definite. Finally, we get a simple estimate for the largest singular value σ (A)ofann n nonsingular complex matrix A. Preliminaries In this section, we review the basic results on matrices needed in this paper. For more comprehensive treatments on matrices we refer to []. Let A be any n n matrix. The Eucledean norm of the matrix A are defined as ( ) / E = a ij () i,j= Also, the spectral norm of the matrix A is = max in λ i, where λ i is eigenvalue of A H A and A H is conjugate transpose of the matrix A. If λ,λ,..., λ n are the eigenvalues of the matrix A, then det A = λ λ...λ n. () The sequare roots of the n eigenvalues of A H A are the singular values of A. Since A H A is Hermitian and positive semidefinite, the singular values of A are real and nonnegative. This let us write them in sorted order σ (A) σ (A)... σ n (A) 0. If σ,σ,..., σ n are the singular values of the matrix A, then E = i= σi (A). (3) Throughout this note, we denote the smallest singular value of A by σ n (A), and its largest singular value by σ (A).
Bounds for singular values of matrices 445 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,..., x n are nonnegative real numbers and λ,λ,..., λ n > 0 with λ i =, then i= n i= x λ i i λ i x i i= and equality holds if and only if x = x =... = x n =. The important unweighted case occurs if we put λ = λ =... = λ n = n : 3 Main Results n x x,...x n x + x,... + x n. (4) n Theorem Let A be any n n positive definite matrix. Then, σ n (A) E. n Proof. From the arithmetic-geometric-mean inequality, we can write ( n ) /n σk (A) n σk (A) (5) and ( n ) /n σk (A) n σ k (A). (6) Threfore, the inequalities (5) and (6) give ( n σ k (A))( Thus, if we consider the identitiy E = we get n ) σk (A) (7) σk (A) and the Ineq. (7), then n E σ k (A).
446 R. Turkmen and H. Civciv Consequently, we have an upper bound for the smallest singular value σ n (A) of the matrix A such that This completes the proof. σ n (A) E. n Theorem Let A be any n-by-n complex matrix. Then, the smallest singular value σ n (A) and the largest singular value σ (A) of A satisfy σ n (A) 4 n (n ) and σ (A) 4 n [ 4 E + E det (I + A A)+ ] /4. Proof. The identity E = σ (A)+... + σ n (A) give 4 E = [ σ (A)+... + σn (A)] 4 [ ] = σk 4 (A)+ σk (A) σ j (A) k>j Thus, from this equality we obtain the inequality 4 E < σk 4 (A)+ σk (A) σ j (A) k>j = σk (A) σm (A) = Hence, the Ineq. (9) implies that m=k [ σk (A) m= = 4 E σk (A) k σm (A) k m= m= σ m (A) ]. (8) σ m (A). (9) n σn 4 (A) k< 4 E. (0)
Bounds for singular values of matrices 447 By solving the Ineq. (0) for σ n (A), we get σ n (A). n (n ) 4 To obtain a lower bound for the largest singular value of A, let us consider the equality (8). Therefore, we write [ ] 4 E σk 4 (A) = σk (A) σj (A). () k>j If we use in () the inequality n ( +σ k (A) ) + σk (A)+ i<jn σ i (A) σ j (A), then we obtain n 4 E + σk (A) ( +σ k (A) ) + σ 4 k (A). () Note that σ k (A), k =,,..., n, are the eigenvalues of A A (with associated eigenvectors x k ). Then, for each j, (I + A A) x j = Ix j + A Ax j = x j + σj (A) x j = ( +σ j (A) ) x j. Therefore, μ k =+σk (A), k =,,..., n, are the eigenvalues of the matrix I + A A. Hence, we can write det (I + A A)= Combining () and (3), we obtain n ( +σ k (A) ). (3) 4 E + E det (I + A A)+ nσ 4 (A). (4) We solve the inequality (4) for σ (A) to obtain σ (A) 4 n [ 4 E + E det (I + A A)+ ] /4.
448 R. Turkmen and H. Civciv Theorem 3 Let A be an n n (n 3) nonsingular complex matrix. Then, the largest singular value of A satisfies ( ) σ (A) (det A) /( n) n n E. n Proof. Using the artihmetic-geometric-mean inequality, we can easily write σ (A) [ σ (A)+... + σ n (A)] ( σ (A)+σ (A)+... + σ n (A) ) = 4 4 E. (5) On the other hand, to obtain an upper bound for the largest singular value of A, we now will apply the artihmetic-geometric-mean inequality on the product σ n 4 (A) det A. Hence, we have σ n 4 (A) (det A) = σ n 4 σ (A) σ (A)...σ n (A) = σ n σ (A)...σn (A) From (5) and (6), we get = ( σ (A)+... + σn (A) ) n n [ σ (σ (A)+... + σ n (A)) ] n. (6) σ n σ n 4 (A) (det A) ( n 4 E 4n 4 ) n. (7) Consequently, from (7) we find an upper bound for the largest singular value of A such that ( ) σ (A) (det A) /( n) n n E. n References [] C. R. Johnson and T. Szulc, Further lower bounds for the smallest singular value, Linear Alg. and Its Appl., 7, 69-79, 998.
Bounds for singular values of matrices 449 [] R. A. Horn, and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, New York, 99, [3] O. Rojo, R. Soto and H. Rojo, Bounds for the spectral radius and the largest singular value, Computers Math. Appl., 36,, 4-50, 998. [4] Yu Yi-Sheng and Gu Dun-he, A note on a lower bound for the smallest singular value, Linear Alg. and Its Appl., 53, 5-38, 997. [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., 43, 4-44, 00. Received: April 7, 007