Lower Bounds for the Spectral Norm - EMIS

Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 6, Issue 4, Article 124, 2005

LOWER BOUNDS FOR THE SPECTRAL NORM JORMA K. MERIKOSKI AND RAVINDER KUMAR D EPARTMENT OF M ATHEMATICS , S TATISTICS AND P HILOSOPHY FI-33014 U NIVERSITY OF TAMPERE , F INLAND [email protected] D EPARTMENT OF M ATHEMATICS DAYALBAGH E DUCATIONAL I NSTITUTE DAYALBAGH , AGRA 282005 U TTAR P RADESH , I NDIA [email protected]

Received 20 December, 2004; accepted 01 September, 2005 Communicated by S. Puntanen

A BSTRACT. Let A be a complex m × n matrix. We find simple and good lower bounds for its spectral norm kAk = max{ kAxk | x ∈ Cn , kxk = 1 } by choosing x smartly. Here k · k applied to a vector denotes the Euclidean norm. Key words and phrases: Spectral norm, Singular values. 2000 Mathematics Subject Classification. 15A60, 15A18.

1. I NTRODUCTION Throughout this paper, A denotes a complex m × n matrix (m, n ≥ 2). We denote by kAk its spectral norm or largest singular value. The singular values of A are square roots of the eigenvalues of A∗ A. Since much is known about bounds for eigenvalues of Hermitian matrices, we may apply this knowledge to A∗ A to find bounds for singular values, but the bounds so obtained are very complicated in general. However, as we will see, we can find simple and good lower bounds for kAk by choosing x smartly in the variational characterization of the largest eigenvalue of A∗ A, (1.1)

kAk = max (x∗ A∗ Ax)1/2 x ∈ Cn , x∗ x = 1 ,

or, equivalently, in the definition (1.10 )

kAk = max kAxk x ∈ Cn , kxk = 1 ,

where k · k applied to a vector denotes the Euclidean norm. ISSN (electronic): 1443-5756 c 2005 Victoria University. All rights reserved.

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J ORMA K. M ERIKOSKI AND R AVINDER K UMAR

Our earlier papers [3] and [4] are based on somewhat similar ideas to find lower bounds for the spread and numerical radius of A. 2. S IMPLE BOUNDS Consider the partition A = (a1 . . . an ) of A into columns. For H ⊆ N = {1, . . . , n}, denote by AH the block of the columns ah with h ∈ H. We accept also the empty block A∅ . Throughout this paper, we let I (6= ∅), K, and L be disjoint subsets of N satisfying N = I ∪ K ∪ L. Since multiplication by permutation matrices does not change singular values, we can reorder the columns, and so we are allowed to assume that A = AI AK AL . Then



 A∗I AI A∗I AK A∗I AL A∗ A =  A∗K AI A∗K AK A∗K AL  . A∗L AI A∗L AK A∗L AL P e , where eh is the h’th standard basis vector of Cn . We denote eH = p h∈H h We choose x = eI |I| in (1.1), where | · | stands for the number of the elements. Then 1 su A∗I AI 2 (2.1) kAk ≥ , |I| where su denotes the sum of the entries. Hence

1

X su A∗I AI 2 1

(2.2) kAk ≥ max = max p ai ,

I6=∅ I6=∅ |I| |I| i∈I and, restricting to I = {1}, . . . , {n}, !1

2

(2.3)

kAk ≥ max kai k = max i

i

X

|aji |2

,

j

and also, restricting to I = N , su A∗I AI (2.4) kAk ≥ n where r1 , . . . , rn are the row sums of A.

12

=

|r1 |2 + · · · + |rn |2 n

12 ,

3. I MPROVED BOUNDS To improve (2.2), choose eI + zeK , x= p |I| + |K| where z ∈ C satisfies |z| = 1. Then 1 2 1 A∗I AI zA∗I AK kAk ≥ p su ∗ ∗ zAK AI AK AK |I| + |K| 

2

2 ! 12

X X X X 1

 =p ai + ak + 2 Re z a∗i ak  ,

|I| + |K| i∈I i∈I k∈K k∈K

J. Inequal. Pure and Appl. Math., 6(4) Art. 124, 2005

http://jipam.vu.edu.au/

L OWER B OUNDS FOR THE S PECTRAL N ORM

a∗i ak 6= 0, and z arbitrary if w = 0, implies 

2

2 21

X 1

X X X ∗   kAk ≥ p a + a + 2 a a .

i k i k

|I| + |K|

which, for z = w/|w| if w =

(3.1)

3

P

i∈I

P

k∈K

i∈I

i∈I k∈K

k∈K

Hence (3.2)



2

2 12

X

X X X 1

 kAk ≥ max p ai + ak + 2 a∗i ak  ,

I6=∅, I∩K=∅ |I| + |K| i∈I

i∈I k∈K

k∈K

and, restricting to I, K = {1}, . . . , {n}, 1 1 (3.3) kAk ≥ √ max kai k2 + kak k2 + 2|a∗i ak | 2 . 2 i6=k It is well-known that the largest eigenvalue of a principal submatrix of a Hermitian matrix is less or equal to that of the original matrix. So, computing the largest eigenvalue of ∗ ai ai a∗i ak a∗k ai a∗k ak improves (3.3) to (

(3.4)

1 kAk ≥ √ max kai k2 + kak k2 + 2 i6=k

h

2

kai k − kak k

2 2

+

i1

2 4|a∗i ak |2

)1 2

.

4. F URTHER IMPROVEMENTS Let B be a Hermitian n × n matrix with largest eigenvalue λ. If 0 6= x ∈ Cn , then x∗ Bx (4.1) λ≥ ∗ . xx We replace x with Bx (assumed nonzero) and ask whether the bound so obtained, (4.2)

λ≥

x∗ B3 x , x∗ B2 x

is better. In other words, is

x∗ B3 x x∗ Bx ≥ x∗ B2 x x∗ x generally valid? The answer is yes if B is nonnegative definite (and Bx 6= 0). In fact, the function x∗ Bp+1 x f (p) = ∗ p (p ≥ 0) xBx is then increasing. We omit the easy proof but note that several interesting questions arise if we instead of nonnegative definiteness assume symmetry and nonnegativity, see [2] and its references. If B = A∗ A, then (4.1) implies kAxk (4.10 ) kAk ≥ , kxk and (4.2) implies a better bound kAA∗ Axk (4.20 ) kAk ≥ . kA∗ Axk



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J ORMA K. M ERIKOSKI AND R AVINDER K UMAR

Since (2.2) is obtained by choosing in (4.10 ) x = eI and maximizing over I, we get a better bound by applying (4.20 ) instead. Because !T X X X A∗ AeI = A∗ ai = a∗1 ai . . . a∗n ai i∈I

i∈I

i∈I

and ! AA∗ AeI =

X

aj a∗j

X

j

ai ,

i∈I

we have (4.3) kAk  

, ! !T

  X X X X X

∗ ∗

ai a a . . . a a I = 6 ∅, a = 6 0 . ≥ max aj a∗j i i i 1 n





j

i∈I

i∈I

i∈I

i∈I

Hence, restricting to I = {1}, . . . , {n} and assuming all the ai ’s nonzero,

(4.4)

! ,

X

kAk ≥ max aj a∗j ai (a∗1 ai . . . a∗n ai )T , i

j

and, restricting to I = N and assuming the row sum vector r = (r1 . . . rn )T nonzero,

! ,

X

(4.5) kAk ≥ aj a∗j r (a∗1 r . . . a∗n r)T .

j

5. E XPERIMENTS We studied our bounds by random matrices of order 10. In the case of (2.2), (3.2), and (4.3), to avoid big complexities, we did not maximize over sets but studied only I = {1, 3, 5, 7, 9}, K = {2, 4, 6, 8, 10}. We considered various types of matrices (positive, normal, etc.). For each type, we performed one hundred experiments and computed means (m) and standard deviations (s) of kAk − bound . bound For positive symmetric matrices, (4.5) was by far the best with very surprising success: m = 0.0000215, s = 0.0000316. For all the remaining types, (4.4) was the best also with surprising success. We mention a few examples. Type Normal Positive Real Complex

m 0.0463 0.0020 0.0261 0.0429

s 0.0227 0.0012 0.0151 0.0162

For positive matrices, also the very simple bound (2.4) was surprisingly good: m = 0.0150, s = 0.0067.



L OWER B OUNDS FOR THE S PECTRAL N ORM

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6. C ONCLUSIONS Our bounds (2.1), (2.3), (2.4), (3.1), (3.3), (3.4), (4.4), and (4.5) have complexity O(n2 ). Also the bounds (2.2), (3.2), and (4.3) have this complexity if we do not include all the subsets of N but only some suitable subsets. Our best O(n2 ) bounds seem to be in general better than all the O(n2 ) bounds we have found from the literature (e.g., the bound of [1, Theorem 3.7.15]). One natural way [5, 6] of finding a lower bound for kAk is to compute the Wolkowicz-Styan [7] lower bound for the largest eigenvalue of A∗ A and to take the square root. The bound so obtained [5, 6] is fairly simple but seems to be in general worse than many of our bounds and has complexity O(n3 ). 7. R EMARK As kAk = kAT k = kA∗ k, all our results remain valid if we take row vectors instead of column vectors, and column sums instead of row sums. We can also do both and choose the better one. R EFERENCES [1] R.A. HORN AND C.R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. [2] H. KANKAANPÄÄ AND J.K. MERIKOSKI, Two inequalities for the sum of elements of a matrix, Linear and Multilinear Algebra, 18 (1985), 9–22. [3] J.K. MERIKOSKI AND R. KUMAR, Characterizations and lower bounds for the spread of a normal matrix, Linear Algebra Appl., 364 (2003), 13–31. [4] J.K. MERIKOSKI AND R. KUMAR, Lower bounds for the numerical radius, Linear Algebra Appl., 410 (2005), 135–142. [5] J.K. MERIKOSKI, H. SARRIA AND P. TARAZAGA, Bounds for singular values using traces, Linear Algebra Appl., 210 (1994), 227–254. [6] O. ROJO, R. SOTO AND H. ROJO, Bounds for the spectral radius and the largest singular value, Comput. Math. Appl., 36 (1998), 41–50. [7] H. WOLKOWICZ AND G.P.H. STYAN, Bounds for eigenvalues using traces, Linear Algebra Appl., 29 (1980), 471–506.

