Estimates of the infinity norms for matrix inverse and the smallest singular value on the GS-SDD matrix*

Zhao Ren-qing，Zhong Zhen-hua，Liu Peng

(School of Mathematics and Statistics，Chuxiong Normal University，Chuxiong，Yunnan，675000)

1.Introduction and notations

Let Mn(C)be the set of all complex matrices of order n×n，A=(aij)∈Mn(C).Denote

N:={1，2，…n}，the set of all indices;

S，a non-empty proper subset of N;

S:=NS，the complement of S;

For the matrix A=(aij)∈Mn(C)，satisfying aii≠0 for all i∈N，we will use the notations

In numerical analysis，a bound is often required for ‖A－1‖∞or the smallest singular value or the condition number κ(A)= ‖A‖·‖A－1‖ for some norm of A ∈ Mn(C).It is known that bounding‖A－1‖ is usually difficult in any norm unless A－1is known explicitly.

For a strictly diagonally dominant matrix，i.e.，

the following well-known bounds were given by Varah in［1］:

In［2］，Johnson obtained the following easily calculable lower bound for σn(A)of an arbitrary matrix A∈Mn(C)，

In［3］，Johnson and Szulc obtained further lower bound for σn(A)of an arbitrary matrix A∈Mn(C)，

Let A=(aij)∈Mn(C)，n≥2，S be a non-empty proper subset of N.We call A an S-SDD if

for all i∈ S，and

In［4］，Nenad Moracˇa obtained the following bound for‖A－1‖∞of an S-SDD matrix for some nonempty proper subset S?N，

In this paper，a new upper bound of‖A－1‖∞and a new lower bound of the smallest singular value are presented.The results obtained are suited to the GS-SDD matrices introduced in section 2，which is an extended matrix class of the strictly diagonally dominant matrices.Examples will show the effectiveness of our new result.

2.Estimates of‖A－1‖∞ and the smallest singular value of A

Lemma 1［5］.A matrix A is an H － matrix if and only if there exists a positive diagonal matrix X such that AX is strictly diagonally dominant.

Definition 1.A matrix A is called a GS-SDD matrix if there exists a positive diagonal matrix D such that AD is an S-SDD matrix.In particular，when D is a unit matrix，AD is an S-SDD matrix.

Lemma 2.Let A=(aij)∈Mn(C)，n≥2，be a matrix with nonzero diagonal entries.If there exists some nonempty proper subset S?N such that

and

then A is an GS-SDD matrix.

Proof Take the positive diagonal matrix D=diag(d1，d2，…，dn)，where

Denote A(1)=AD=()，we have

and for each i∈N，

then A(1)=AD is an S-SDD matrix from(9)and(10)，thus A is an GS-SDD matrix.

Lemma 3.Let A=(aij)∈Mn(C)，n≥2，be a matrix with nonzero diagonal entries.If there exists some nonempty proper subset S?N such that(9)and(10)hold，then A is an H －matrix.

Proof Denote

(1)For i∈ S，

(2)For j∈S，

Hence，

Then，A is a nonsingular H －matrix.

For convenience，we denote

Theorem 1.Let A=(aij)∈Mn(C)，n≥2，be a matrix with nonzero diagonal entries.If there exists some nonempty proper subset S?N such that

and

then

Proof From the proof of Lemma 1，we have that A(1)=AD is an S-SDD matrix.By(8)we obtain

From A(1)=AD，we get that

Similarly，we have the following result.

Corollary 1.Let A=(aij)∈Mn(C)，n≥2，be a matrix with nonzero diagonal entries.If there exists some nonempty proper subset S?N such that

and

then

By Theorem 1 and Corollary 1，we have the following lower bound of σn(A)，immediately.

Theorem 2.Let A=(aij)∈Mn(C)，n≥2，be a matrix with nonzero diagonal entries.If there exists some nonempty proper subset S?N such that

and

then

Proof Because the inequality‖A‖22≤‖A‖∞·‖A‖1holds for every A∈Mn(C)，by Theorem 1 and Corollary 1，the conclusion is obtained immediately

3.Examples

Example 1.Let

It is easy to verify that A is a GS-SDD matrix.Taking S={1，2}，by(11)of Theorem 1，we have

Taking S={1，2}，by(12)of Corollary 1，we have

And by(13)of Theorem 2，σn(A)≥0.5773(the true value is σn(A)=3.9438).However，A is not a diagonally dominant matrix，the estimates(1)and(2)cannot be used.Using the estimates(3)，(4)and(5)，we get the following bounds，respectively:

σn(A)≥0.1980(by(3)of［2］);σn(A)≥0.5131(by(4)of［3］);σn(A)≥0(by(5)of［3］).

Example 2.Let

It is easy to verify that A is a GS-SDD matrix.Taking S={1，2}，by(11)of Theorem 1，we have

Taking S={1，2}，by(12)of Corollary 1，we have

And by(13)of Theorem 2，σn(A)≥0.24(the true value is σn(A)=5.3586).However，A is not a diagonally dominant matrix，the estimates(1)and(2)cannot be used.Using the estimates(3)，(4)and(5)，we get the following meaningless bounds(negative lower bounds)，respectively:

σn(A)≥－0.5(by(3)of［2］);σn(A)≥－0.4438(by(4)of［3］);σn(A)≥－0.2485(by(5)of［3］).

The two examples show that the estimate formulas(11)in Theorem 1 and(12)of Corollary 1 are effective and that estimate formula of Theorem 2 is much better than those in［2］and［3］.

［1］J.M.Varah.A lower bound for the smallest singular value.Linear Algebra Appl.1975(11):3—5.

［2］C.Johnson.A Ger?gorin-type lower bound for the smallest singular value.Linear Algebra Appl.1989(112):1—7.

［3］C.R.Johnson，T.Szulc.Further lower bounds for the smallest singular value.Linear Algebra Appl.1998(272):169—179.

［5］Ting Zhu Huang.Estimation of‖A－1‖∞and the smallest singular value.Comput.Math.Appl.2008(55):1075—1080.