Selçuk J. Appl. Math. Selçuk Journal of Vol. 6. No.1. pp. 21-28, 2005 Applied Mathematics
Bounds for the Product of Singular Values A. Dilek Güngör
Department of Mathematics Art and Science Faculty Selçuk University, 42031 Konya, Turkey;
e-mail: agungor@ selcuk.edu.tr
Received :February 2, 2005
Summary. Let A be a n n complex matrix with 1 > 2 > ::: > n and
let 1 6 k < l 6 n. Bounds for 1::: k, k::: l and n k+1::: n , involving
k; n; ri(ci); i and det A where ri(ci) is the Euclidean norm of the i-th row
(column) of A and i’s are positive real numbers such that 21+ 22+:::+ 2n= n,
are presented.
Keywords: Singular value, Frobenius norm, row Euclidean norm, column Euclidean norm.
1. Introduction and Preliminaries
Let A be a n n complex matrix.Let i(A)’s be the singular values ofA such
that
1(A)> 2(A)> ::: > n(A) :
It is well known that
(1) 21(A) + 22(A) + ::: + 2n(A) = kAk2F and
(2) 1(A) 2(A) ::: n(A) = jdet Aj
where kAkF and det A denote the Frobenius norm of A and the determinant of
Wang and Zhang [5] have established some inequalities for the eigenvalues of the product of positive semide…nite Hermitian matrices. Merikoski and Virtanen [3] obtained bounds for the eigenvalues using the trace and determinant. Rojo [4] de…ned B matrix.
In Section 2, we have obtained bounds for products of singular values using the bounds for the eigenvalues in [3] and the inequalities (1) and (2). In Section 3, we have bounds for the products of singular values using row (column) norm and determinant [1].
Firstly, we give some preliminaries required for our study. We de…ne (3) 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
(4) 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 B = AD,
(5) D = diag 1 c1(A) ; 2 c2(A) ; :::; n cn(A) ;
where ci(A) is the Euclidean norm of the i-th column of A. Again, kBkF =pn
and if 1 = 2 = :: = n = 1 then each column of B is a unit vector in the
Euclidean norm.
Theorem 1 [5] Let A; B 2 Cn n and let 1 i1< ::: < ik n: Then
(6) k Y t=1 it(AB)6 k Y t=1 it(A) t(B) and (7) k Y t=1 t(AB)> k Y t=1 it(A) n it+1(B) :
Theorem 2 [2] Let A 2 Cn n; B 2 Cn nand let 1 i
(8) k Y t=1 it(AB)> k Y t=1 it(A) n t+1(B) :
2. Bounds For Product of Singular Values Using Norm and Determi-nant
We obtained some results for the singular values by using the results given in [3] where A be a square matrix with singular values
1> 2> ::: > n> 0:
Corollary 1. Let 1 k n 2. Then
(9) 1::: k6 2 6 4(det A)1 2 0 @ 1 n k kAk2F k + 1 !k+11 A n k3 7 5 1 2(n k 1) Let 26 k 6 n 1. Then (10) n k+1::: n> 2 4k (det A)2 n k + 1 kAk2E !n k+13 5 k 2(k 1)
Corollary 2. Let 1 k l n 2. Then
(11) 8 > < > :jdet Aj 2(1 k n)l+2 2 4(n k + 1) k kAk2E !k3 5 n k+19> = > ; 1 2(n k) 6 k::: l 6 ( 1 jdet Aj 2[(n l)k+l k+l] n 1 n l kAk2 E l+1 l+1 n l) 1 2(n l 1) and (12) (n k + 1) (det A)2 k kAk2 E k 2(nl k+1k) 6 k::: l 6 ( 1 (det A)2 n l1 kAk 2 E l+1 l+1 n l) l k+1 2l(n l 1) :
3. Bounds For Product of Singular Values Using Row (Column) Norm and Determinant
Since the matrices P A, AP and A have the same singular values for any permutation matrix P , we assume for this section, without loss of generality, that the rows and columns of A are such that
(13) r1(A)6 r2(A)6 ::: 6 rn(A)
(14) c1(A)6 c2(A)6 ::: 6 cn(A)
and let
(15) 0 < n 6 ::: 6 26 1
for i’s as in (4).
Theorem 3Let 1 k < l n 2. Then
(16) 1::: l k+1> l Y i=k ri i 8 < :(n k + 1) n Y i=1 i ri det A !2 k n k9= ; l k+1 2(n k)
where ri’s and i’s be as in (13) and (15), respectively.
Corollary 3. Let 1 k n 2. Then
Proof . Write inequality (12) for the matrix B which is de…ned in Section 1. Taking i1= k, ik = l in (6) and applying B = DA which is de…ned in Section
1 completes the proof.
Theorem 4Let 1 k < l n 2. Then
n l+k::: n 6 l Q i=k ri i 8 > > > > < > > > > : 0 B B @ n Q i=1 ri i jdet Aj 1 C C A 2[(n l)k+l k+1] n (17) " 1 n l n l + 1 l+1#n l9= ; 1 2(n l 1)
where ri’s and i’s be as in (13) and (15), respectively.
Proof. To prove inequlity (17), we write (11) for the matrix B which is de…ned in Section 1. Taking i1= k , ik = l in (8) and using B = DA we get the
inequality (17)
We can establish di¤erent inequalities for the product of singular values using the matrices A = D 1B and A = BD 1which can be obtained from B = DA
and B = AD. In this case, we will suppose that the rows and columns of A are such that
(18) r1(A)> r2(A)> ::: > rn(A)
(19) c1(A)> c2(A)> ::: > cn(A)
and
(20) 0 < 16 26 ::: 6 n
for 0
i s in (4).
Let A be a square matrix with singular values
1> 2> ::: > n> 0:
Therefore we can give the following theorems. Theorem 5Let 1 < k n 1. Then
1::: k > n Y i=n k+1 max (ri; ci) i 8 < :k n Y i=1 i min (ri; ci)jdet Aj !2 (21) n k + 1 n n k+1)2(kk1)
where ri(ci)’s and i’s be as in (13) (14) and (15), respectively.
Proof. Firstly, write inequality (10) for the matrix B which is de…ned in Section 1. Taking i1= 1 and ik = k in (7) and applying the matrix A = D 1B,
we have (21).
(22) n Y i=n l+k ci i 8 < :(n k + 1) n Y i=1 i ci jdet Aj !2 k n k9= ; l k+1 2(n k) 6 k::: l 6 l k+1Y i=1 ci i 8 > > > < > > > : 0 B B @ n Q i=1 ci i jdet Aj 1 C C A 2 " 1 n l n l + 1 l+1#n l 9 > > > = > > > ; l k+1 2l(n l 1)
where ci’s and i’s be as in (14) and (15), respectively.
Proof. We write inequality (11) for the matrix B which is de…ned in Section 1. Taking i1= k, ik = l in (6) and applying A = BD 1 matrix, we have (??).
Theorem 7Let 1 k n 1. Then
(23) n Q i=n k+1 ri i ( k n Q i=1 i ri jdet Aj 2 n k+1 n n k+1 ) k 2(k 1) 6 n k+1::: n 6 n Y i=n k+1 ri i 8 > > > < > > > : 0 B B @ n Q i=1 ri i det A 1 C C A 2 " 1 n k n k + 1 k+1#n k 9 > > > = > > > ; 1 2(n k 1)
where ri’s and i’s be as in (13) and (15), respectively.
Proof. Firstly, write the inequalities (9) and (10) for the matrix B which is de…ned in Section 1. Taking i1= n k + 1, ik= n in (6) and (8) and applying
A = D 1B matrix, we have (23). Example 1 Let A = 2 6 6 4 1 2 1 2 1 1 2 1 3 2 1 1 1 3 2 2 3 7 7
5. Singular values of the matrix A are
1= 6:673; 2= 6:673; 3= 6:673; and 4= 6:673: In the following table the
best bound are underlined.
knl 1 2 3 4
1 1 1 2 1 2 3 1 2 3 4
2 2 2 3 2 3 4
3 3 3 4
4 4
1.The product of singular values: knl 1 2 3 4 1 6:673 13:008 15:362 8 2 1:949 2:302 1:199 3 1:181 0:615 4 0:521
2. Lower bounds for the product of singular values:
(24) kn l 1 2 3 4 1 0:198 0:415 2:063 (25) kn l 1 2 3 4 1 0:025 0:49 2:861 (26) knl 1 2 3 4 1 0:445 0:377 2:017 2 0:097 0:038 3 0:002 (27) knl 1 2 3 4 1 2 3 0:002
3. Upper bounds for the product of singular values:
(28) knl 1 2 3 4 1 2 3 404:69 (29) kn l 1 2 1 5277:946
The bounds for individual singularvalues are found from the diagonals of these tables. We have seen that some bounds for 1 = 2 = :: = n = 1 are better
than those for i’s (i = 1; :::; n) such that 21+ 22+ ::: + 2n = n while some
References
1. Güngör A. D. (2004): Bounds For Singular and Norm Values , PhD Thesis, Selçuk University, Konya.
2. Marshall, A. W., Olkin, I.(1979): Inequalities, Theory of Majorization and its Applications, Academic, New York.
3. Merikoski, J. K., Virtanen, A. (1997): Bounds for eigenvalues using the trace and determinant, Linear Algebra and its Applications 264, 101-108.
4. Rojo, O. (1999): Further bounds for the smallest singular value and the spectral condition number, Computers and Mathematics with Applications, Vol. 38, No: 7-8, 215 - 228.
5. Wang, B., Zhang, F. (1992): Some inequalities for the eigenvalues of the product of positive semide…nite Hermitian matrices, Linear Algebra and its Applications 160, 113-118.
