Selçuk J. Appl. Math. Selçuk Journal of Vol. 10. No. 2. pp. 43-51, 2009 Applied Mathematics
Bounding the extremal eigenvalues of a complex matrix Ay¸se Dilek Güngör
Department of Mathematics Art and Science Faculty Selçuk University 42031 (Cam-pus) Konya, Türkiye
e-mail: agungor@ selcuk.edu.tr
Received Date: November 24, 2008 Accepted Date: November 17, 2009
Abstract. In this study, we have obtained bounds for extreme singular values of a complex matrix A of order n n.
In addition, we have found bounds for the extreme singular values of Hilbert matrix, its Hadamard square root, Cauchy-Hankel matrix in the forms H = (1= (i + j 1))ni;j=1, H 1=2 = (1= (i + j 1)1=2
)n
i;j=1 and Hn =
[1=(g + (i + j)h)]ni;j=1; respectively.
Key words: Hilbert matrix; Hadamard square root; Cauchy-Hankel matrix; Singular value.
2000 Mathematics Subject Classi…cation: 15A18, 15A60, 15A15. 1.Introduction and Preliminaries
Let A = (aij) be n n matrix with nonzero entries. Then the Hadamard
inverse of A given by A ( 1) = (1=a
ij)ni;j=1 . Let B = (bij) be n n matrix
with all nonnegative entries. Then the Hadamard square root of B is given by B 1=2= (b1=2
ij )ni;j=1 [5].
The matrix
(1.1) H = (1=(i + j 1))ni;j=1
is well known as Hilbert matrix. Hence the Hadamard square root of Hilbert matrix, denoted by
(1.2) H 1=2= (1=(i + j 1)1=2)ni;j=1:
Let C = [1=(xi yj)]ni;j=1(xi 6= yj) be general Cauchy matrix and Hn =
[hi+j]n 1i;j=0 be general Hankel matrix. Every n n Cauchy-Hankel matrix is
(1.3) Hn=
1 g + (i + j)h
n
i;j=1
where h 6= 0; g and h are any numbers and g=h is not integer. Hankel matrices are symmetric.
Now, we give some preliminaries related to our study. Let A be an n n complex matrix. Let i(A) values (i = 1; :::; n) such that 1(A) 2(A) ::: n(A) be the singular values of A . It is well known
that
1(A) 2(A) ::: n(A) = jdet Aj
where det A is determinant of A:
In [1], Euler introduced the gamma function ; de…ned for Rez > 0 by the improper integral (z) = 1 Z 0 uz 1e udu:
Actually this function can be extended to the rest of the complex plane, ex-cepting the non-positive integers. Meanwhile the logarithmic derivative of the gamma function is known as the psi or digamma function ([4]), that is, it is presented by (x) = d dxln = 0 (x) (x)
for any complex x not equal to a negative integer. In fact the derivatives (m; x) = d dxm (x) = d dxm d dxln
of the digamma function are called polygamma functions.The equation
(n + x) (x) = n 1 X r=0 1 x + r
is well known for the psi function, and so one can easily obtain an equation for the polygamma function by applying di¤erential:
(1; x) (1; n + x) = n 1 X r=0 1 (x + r)2:
To minimize the numerical round-o¤ errors in solving the system Ax = b; it is normally convenient that the rows of A be properly scaled before the solution procedure begins. One way is to premultiply by the diagonal matrix
(1.4) D=diag 1 r1(A) ; 2 r2(A) ; :::; n rn(A) ;
where ri(A) is the Euclidean norm of the i th row of A and 1; 2; :::; n are
positive real numbers such that
(1.5) 21+ 22+ ::: + 2n = n:
Clearly, the Euclidean norm of the coe¢ cient matrix B = DA of the scaled system is equal topn and if 1 = 2 = ::: = n = 1 then each row of B is a
unit vector in the Euclidean norm. Also, we can de…ne B0 = AD0,
(1.6) D0=diag 1 c1(A) ; 2 c2(A) ; :::; n cn(A) ;
where ci(A) is the Euclidean norm of the i th columm of A: Again, k B
0
kE =
p
n and if 1 = 2 = ::: = n = 1 then each column of B
0
is a unit vector in the Euclidean norm.
Since the matrices P A; AP and A have the same singular values for any permutation matrix P , we assume, without loss of generality, that the rows and columns of A are such that
(1.7) r1(A) r2(A) ::: rn(A) ,
(1.8) c1(A) c2(A) ::: cn(A) ,
and i values in (1:5) are ordered in such a way that
(1.9) 0 < 1 2 ::: n.
Estimating the extreme singular values have theoretical and practical interests. In fact there exists a vast literature that studies estimating the extreme singular values ([3]). Also, in [7], Türkmen et. al. gave two upper bounds of n(A)
showing that if A 2 Cn n is a nonsingular complex matrix, then
(1.10) n(A) kAkp F
n and
(1.11) n(A) p4 kAkF
n (n 1):
In addition to these above, in [2], the best possible upper bound for n(A) has
been obtained by Güngör et. al.
After all, in this paper, we obtain some new upper bounds which improve the results obtained in the previous papers for n(A) and lower bounds for 1(A)
using the matrix B is de…ned in [6] (see Remark 1 below). In addition, we have established the di¤erent upper and lower bounds for extreme singular values of
the Hilbert matrix and its Hadamard square root and Cauchy-Hankel matrix. Finally, we have given numerical results related to these bounds.
Theorem 1. [8] Assume that A and B are two arbitrary n n matrices with the singular values
n(A) ::: 1(A) ; n(B) ::: 1(B) .
Then the singular values
n(AB) ::: 1(AB)
of the matrix AB satisfy
(a) i(AB) = i i(A) = i i(B) ; n(B) i 1(B) ; n(A) i 1(A) ; 1 i n; (b) i(AB) = wi p i(A) i(B); p n(A) n(B) wi p 1(A) 1(B) ; 1 i n:
2. Bounds for extreme singular values
Theorem 2. Let A be an n n non-singular complex matrix. Let ri(A) values,
ci(A)’values and i’values be as in (1:7) ; (1:8) and (1:9) ; respectively. Then,
(2.1) n(A) n maxfrn; cng n v u u u t jdet Aj n Q i=1 i minfri;cig and (2.2) 1 minfr1; c1g n v u u u t jdet Aj n Q i=1 i maxfri;cig 1(A) :
Proof. Firstly, we can write Theorem1 (a) in the form
i(AB) = i i(A) = i i(B)
where n(B) i 1(B) ; n(A) i 1(A) ; 1 i n: Hence, we
obtain that (2.3) n(B) i (AB) i(A) 1 (B) and (2.4) n(A) i (AB) i(B) 1(A) :
From (2.3) and (2.4) , we write (2.5) n(B) 1(A) i(B) i(A) 1(B) n(A) :
By applying B = DA and B0 = AD0 matrices to (2.5) and taking product on i;we obtain the theorem.
If we put 1= 2= ::: = n = 1 in (2.1) and (2.2), we get Corollary3.
Corollary 3. Let A be an n n non-singular complex matrix. Then,
(2.6) n(A) n s jdet Aj n Q i=1minfri ; cig maxfrn; cng and (2.7) n s jdet Aj n Q i=1maxfr i; cig minfr1; c1g 1 (A) :
3. Bounds for extreme singular values of Hilbert matrix , its Hadamard square root and Cauchy-Hankel matrices
Theorem 4. Let H; ri(A) values , ci(A)’ values and i values be as in
(1:1) ; (1:7) ; (1:8) and (1:9) ;respectively. Then
(3.1) n(H) n a2 n v u u u t jdet Hj n Q i=1 i a3 and (3.2) 1 a1 n s jdet Hj a3 1 (H) where a1= r (1; n + 1) + 2 6 ; a2= p (1; 2n) + (1; n) and a3= p (1; i + n) + (1; i):
Proof. For the matrix H in (1:1) we have r1(H) = c1(H) = r (1; n + 1) + 2 6 ; rn(H) = cn(H) = p (1; 2n) + (1; n) and n Y i=1 minfri; cig = n Y i=1 maxfri; cig = n Y i=1 p (1; i + n) + (1; i):
By applying this equalities to (2:1) and (2:2), we obtain (3:1) and (3:2). Hence the result.
We note that the Euler-Mascheroni constant is de…ned by
= lim n!1 n X k=1 1 k ln (n) ! = 0:577215664901533:::. and this will be used in the following result:
Theorem 5. Let the matrix H 1=2 and the values ri(A) ; ci(A)’; i be as in
(1:2) ; (1:7) ; (1:8) and (1:9) ;respectively. Then
(3.3) n H 1=2 n b2 n v u u u t det H 1=2 n Q i=1 i b3 and (3.4) 1 b1 n s det H 1=2 b3 1 H 1=2 ; where b1= p (n + 1) + ; b2= p (2n) (n) and b3= p (i + n) (i):
Proof. For the matrix H 1=2 in (1:2) we have r1 H 1=2 = c1 H 1=2 = p (n + 1) + ; rn H 1=2 = cn H 1=2 = p (2n) (n)
and n Y i=1 minfri; cig = n Y i=1 maxfri; cig = n Y i=1 p (i + n) (i): By applying these equalities to Theorem 2, we obtain (3:3) and (3:4).
Theorem 6. Let Hn; ri(A) values, ci(A) values and i’values be as in
(1:3) ; (1:7) ; (1:8) and (1:9) ;respectively. Then
(3.5) n(Hn) n c2 n v u u u t jdet Hnj n Q i=1 i c3 and (3.6) 1 c1 n s jdet Hnj c3 1(Hn) where c1= s 1; n +5 2 40 9 + 2 2 ; c2= s 1; 2n +3 2 + 1; n + 3 2 and c3= s 1; n +3 2 + i + 1; 3 2+ i :
Proof. For the matrix Hn in (1:3) we have
r1(Hn) = c1(Hn) = s 1; n +5 2 40 9 + 2 2 ; rn(Hn) = cn(Hn) = s 1; 2n +3 2 + 1; n + 3 2 and n Y i=1 minfri; cig = n Y i=1 maxfri; cig = n Y i=1 s 1; n +3 2 + i + 1; 3 2+ i : By applying these equalities to (2:1) and (2:2), we obtain (3:5) and (3:6).
4. Numerical Results
We will take 1= 2= ::: = n = 1 and n = 5 in the following examples.
Example 7. Let x1= n s jdet Hj n Q i=1 a3 a2 and x2 = n s jdet Hj n Q i=1 a3 a1
where a1; a2and a3are as in Theorem 4. For extreme singular values of the
matrix H in (1:1), we have the following values:
n(H) x1 x2 1(H) 0:6869025215:10 5 0:008676528536 0:002444303523 1:567050691 Example 8. Let x3= n s det H 1=2 Qn i=1 b3 b2 and x4= n s det H 1=2 Qn i=1 b3 b1 :
where b1; b2 and b3are as in Theorem5. For extreme singular values of the
matrix H 1=2 in (1:2), we have the following values:
n H 1=2 x3 x4 1 H 1=2 0:8127124207:10 5 0:01176332609 0:006722156165 2:533602599 Example 9. Let x5= n s jdet Hnj n Q i=1 c3 c2 and x6 = n s jdet Hnj n Q i=1 c3 c1
where c1; c2 and c3are as in Theorem6. For extreme singular values of the
matrix Hn in (1:3), we have the following values:
n(Hn) x5 x6 1(Hn)
0:8421550433:10 5I 0:001529009759 0:0007121582225 0:928688586
Remark 10. As we depicted in the begining of this paper, there have been obtained some results on the bounds that studied in here. For instance in Example 7 above, while we did obtain 0; 008676528536 for the smallest singular value of the matrix H given in (1:1), in [2] the same value has been obtained as 0; 4861024028. (Also for Examples 8 and 9, similar di¤erences has been obtained other than [2]). Further, in [7], the auhors obtained 0; 7070027739 and 0; 7475644164 for the smallest singular value of the matrix H in (1:1) by using equations (1:10) and (1:11), respectively. But it is seen quite clearly that our bounds given in the table of Example 7 are much better than these. So all these notes show that our results in this paper are more valuable than the others. References
1. Euler L., De progressionibus transcendentibus seu quarum termini generales alge-braice dari nequeunt, in: C. Bohm, G. Faber (Eds.) Opera Omnia (1) vol. 14, B. G. Teubner, Berlin, 1925, p. 1-24.
2. Güngör A. D. and Türkmen R., Bounds for extreme singular values of a complex matrix and its applications, Mathematical Inequalities & Applications, Vol. 9, No. 1 (2006), 23-31.
3. Horn R. and Johnson C. R., Matrix Analysis, Cambridge University Press, New York, 1991.
4. Moenck R., On computing closed forms for summations, in: Proceedings of MAC-SYMA users’conference, Berkley, 1977, p. 225-236.
5. Reams R., Hadamard inverses, square roots and product of almost semide…nite matrices, Linear Algebra and Its Applications, 288 (2000), 35-43.
6. Rojo O., Further bounds for the smallest singular value and the spectral condition number, Computers and Mathematics with Applications, Vol. 38, No. 7-8, (1999) 215 – 228.
7. Türkmen R. and Civciv H., Some bounds for the singular values of matrices, Applied Mathematical Sciences, 49 (1) (2007) 2443-2449.
8. Yongzhong S., Estimations for the eigenvalues of the product of matrices (Chinese), J. Nanjing Norm. Univ. Nat. Sci. Ed. 16(2) (1994) 10-13.
