In this section, we give two new lower bounds for which improve the ones in [8] and [9].
Lemma 3.1 If and is a doubly stochastic matrix, then
Proof This proof is similar to the ones of Lemma 3.2 in [8] and Theorem 3.2 in [9]. □
Theorem 3.1 Let and be a doubly stochastic matrix. Then
Proof Firstly, we assume that A is irreducible. By Lemma 2.5, we have
Denote
Since A is an irreducible matrix, we know that . So, by Lemma 2.4, there exists such that
or equivalently,
Secondly, if A is reducible, without loss of generality, we may assume that A has the following block upper triangular form:
where is an irreducible diagonal block matrix, . Obviously, . Thus the reducible case is converted into the irreducible case. This proof is completed. □
Theorem 3.2 If is a strictly diagonally dominant by row, then
Proof Since A is strictly diagonally dominant by row, for any , we have
or equivalently,
(1)
So, we can obtain
(2)
and
Therefore, it is easy to obtain that
Obviously, we have the desired result
This proof is completed. □
Theorem 3.3 If is strictly diagonally dominant by row, then
Proof Since A is strictly diagonally dominant by row, for any , we have
i.e.,
(3)
So, we can obtain
(4)
and
Therefore, it is easy to obtain that
Obviously, we have the desired result
□
Remark 3.1 According to inequalities (1) and (3), it is easy to know that
and
That is to say, the result of Lemma 2.2 is sharper than the ones of Theorem 2.1 in [8] and Lemma 2.2 in [9]. Moreover, the results of Theorem 3.2 and Theorem 3.3 are sharper than the ones of Theorem 3.1 in [8] and Theorem 3.3 in [9], respectively.
Theorem 3.4 If is strictly diagonally dominant by row, then
Proof This proof is similar to the one of Theorem 3.5 in [8]. □
Remark 3.2 According to inequalities (2) and (4), we get
and
That is to say, the bound of Theorem 3.4 is sharper than the ones of Theorem 3.5 in [8] and Theorem 3.4 in [9], respectively.
Remark 3.3 Using the above similar ideas, we can obtain similar inequalities of the strictly diagonally M-matrix by column.