Some Bounds for the Singular Values of Matrices

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1 Applied Mathematical Sciences, Vol., 007, no. 49, 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 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

2 444 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).

3 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).

4 446 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)

5 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.

6 448 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.

7 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