On improvements of Fischer’s inequality and Hadamard’s inequality for $K_0$-matrices

Abstract

In this paper, the class of $K_0$-matrices, which includes positive definite matrices, totally positive matrices, M-matrices and inverse M-matrices, is first introduced and the refinements of Fischer’s inequality and Hadamard’s inequality for $K_0$-matrices are obtained. Some previous well-known results for totally nonnegative matrices can be regarded as the special case of this paper.

MSC:15A15.

1 Introduction

All matrices considered in this paper are real. For an $n\times n$ matrix $A,\alpha ,\beta \subseteq 〈n〉=\{1,2,\dots ,n\}$, the submatrix of A lying in rows indexed by α and the columns indexed by β will be denoted by $A[\alpha ,\beta ]$. If $\alpha =\beta$, then the principal submatrix $A[\alpha ,\alpha ]$ is abbreviated to $A[\alpha ]$. For any $\alpha \subseteq 〈n〉$, let ${\alpha }^c$ denote the complement of α relative to $〈n〉$, and let $|\alpha |$ denote the cardinality of α. If $\alpha =\mathrm{\varnothing }$, we define $detA[\mathrm{\varnothing }]=1$. We use $S_n$ for the symmetric group on $〈n〉$.

An $n\times n$ matrix A is called a $P_0$-matrix (P-matrix) if all the principal minors of A are nonnegative (positive). A P-matrix A is called 1-minor symmetric if $detA[\alpha ,\beta ]\cdot detA[\beta ,\alpha ]\ge 0$, whenever $|\alpha |=|\beta |=1+|\alpha \cap \beta |$. Of course, each of the P-matrices, such as positive definite matrices (PD), totally positive matrices (TP), M-matrices (M) and inverse M-matrices ($M^{-1}$), is 1-minor symmetric, see [1]. We have known that the following multiplicative principal minor inequalities are classical for PD, M and $M^{-1}$ matrices:

• Hadamard: $detA\le {\prod }_{i=1}^n{a}_{ii}$;

• Fischer: $detA\le detA[S]\cdot detA[S^c]$, for $\mathrm{\forall }S\subseteq 〈n〉$;

• Koteljanskii: $detA[S\cup T]\cdot detA[S\cap T]\le detA[S]\cdot detA[T]$, for $\mathrm{\forall }S,T\subseteq 〈n〉$.

These inequalities also hold for totally nonnegative matrices (TN), see [1–6].

The study of multiplicative principal minor inequalities has been actively going on for many years, many authors have done various wonderful works on this topic, see [7–11]. In [11], Zhang and Yang improved Hadamard’s inequality for totally nonnegative matrices as follows:

If $A=[a_{ij}]$ is an $n\times n$ totally nonnegative matrix with ${\prod }_{i=1}^n{a}_{ii}>0$, then

$$detA\le min\{\prod _{i=1}^n{a}_{ii}-\underset{1\ne \sigma \in S_n}{max}{\left(\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right)}^{1/2},\underset{k\in 〈n〉}{min}{a}_{kk}\underset{i\ne k}{\overset{n}{\prod _{i=1}}}(a_{ii}-\frac{a_{ik}{a}_{ki}}{a_{kk}})\}.$$

(1.1)

Hadamard’s inequality for some subclasses of $P_0$-matrices is an important inequality in matrix analysis, inequality (1.1) is the generalization of Hadamard’s inequality for totally nonnegative matrices. It is a noticeable problem to generalize inequality (1.1) for totally nonnegative matrices to other classes of matrices. In this paper we give some new upper bounds of Fischer’s inequality and Hadamard’s inequality for a subclass of $P_0$-matrices and extend the corresponding results due to Zhang and Yang (see [11]).

2 Some lemmas

To avoid triviality, we always assume $n>1$. We will need important Sylvester’s identity for determinants (see [12]).

Lemma 2.1 [12]

Let A be an $n\times n$ matrix, $\alpha \subseteq 〈n〉$, and suppose $|\alpha |=k$ ($1\le k\le n-1$). Define the $(n-k)\times (n-k)$ matrix $B=(b_{ij})$, with $i,j\in {\alpha }^c$, by setting $b_{ij}=detA[\alpha \cup \{i\},\alpha \cup \{j\}]$ for every $i,j\in {\alpha }^c$. B is called the Sylvester matrix of A associated with $A[\alpha ]$. Then Sylvester’s identity states that for each $\delta ,\gamma \subseteq {\alpha }^c$, with $|\delta |=|\gamma |=l$,

$$detB[\delta ,\gamma ]={(detA[\alpha ])}^{l-1}detA[\alpha \cup \delta ,\alpha \cup \gamma ].$$

(2.1)

For convenience, we introduce the following definition.

Definition 2.1 A $P_0$-matrix (P-matrix) A is called a $K_0$-matrix (K-matrix) if every principal submatrix of A satisfies Koteljanskii’s inequality.

Obviously, each principal submatrix of a $K_0$-matrix (K-matrix) is a $K_0$-matrix (K-matrix). Of course, each of the matrices PD, TP, M and $M^{-1}$ is a K-matrix, the totally nonnegative matrices are $K_0$-matrices. In fact, an evident necessary and sufficient condition for a K-matrix was given in [1].

Lemma 2.2 [1]

A P-matrix satisfies Koteljanskii’s inequality if and only if it is 1-minor symmetric.

There are K-matrices that lie in none of the classes PD, TP, M and $M^{-1}$, e.g.,

$$\left(\begin{array}{ccc}4& 2& -3\\ 1& 3& 1\\ -1& 1& 2\end{array}\right).$$

Lemma 2.3 If an $n\times n$ matrix $A=[a_{ij}]$ is a $K_0$-matrix, then $a_{ij}{a}_{ji}\ge 0$ for $\mathrm{\forall }i,j\in 〈n〉$.

Proof For $\mathrm{\forall }i,j\in 〈n〉$, $i<j$, we consider the submatrix $A[\{i,j\}]$ of the $K_0$-matrix A. Since $A[\{i,j\}]$ is a $K_0$-matrix, by the Hadamard’s inequality, we have

$$detA[\{i,j\}]=a_{ii}{a}_{jj}-a_{ij}{a}_{ji}\le a_{ii}{a}_{jj},$$

therefore

$$a_{ij}{a}_{ji}\ge 0\text{for }\mathrm{\forall }i,j\in 〈n〉.$$

This completes the proof. □

Lemma 2.4 Let A be a K-matrix, B be the Sylvester matrix of A associated with $A[\alpha ]$, then B is a K-matrix.

Proof For $\mathrm{\forall }\delta \subseteq {\alpha }^c$, with $|\delta |=l$, by Sylvester’s identity (2.1), we have

$$detB[\delta ,\delta ]={(detA[\alpha ])}^{l-1}detA[\alpha \cup \delta ,\alpha \cup \delta ]>0,$$

this means that B is a P-matrix.

For $\mathrm{\forall }{\delta }^{\prime }$, $\gamma \subseteq {\alpha }^c$, with $|{\delta }^{\prime }|=|\gamma |=1+|{\delta }^{\prime }\cap \gamma |=k_1$, by Sylvester’s identity (2.1), we have

$$\begin{array}{r}detB[{\delta }^{\prime },\gamma ]detB[\gamma ,{\delta }^{\prime }]\\ ={(detA[\alpha ])}^{2k_1-2}detA[\alpha \cup {\delta }^{\prime },\alpha \cup \gamma ]detA[\alpha \cup \gamma ,\alpha \cup {\delta }^{\prime }]>0.\end{array}$$

(2.2)

Obviously,

$$\begin{array}{c}\begin{array}{rl}|\alpha \cup {\delta }^{\prime }|& =|\alpha |+\left|{\delta }^{\prime }\right|=1+|\alpha |+|{\delta }^{\prime }\cap \gamma |\\ =1+|(\alpha \cup {\delta }^{\prime })\cap (\alpha \cup \gamma )|,\end{array}\hfill \\ \begin{array}{rl}|\alpha \cup \gamma |& =|\alpha |+|\gamma |=1+|\alpha |+|{\delta }^{\prime }\cap \gamma |\\ =1+|(\alpha \cup {\delta }^{\prime })\cap (\alpha \cup \gamma )|.\end{array}\hfill \end{array}$$

That is,

$$|\alpha \cup {\delta }^{\prime }|=|\alpha \cup \gamma |=1+|(\alpha \cup {\delta }^{\prime })\cap (\alpha \cup \gamma )|.$$

By Lemma 2.2 and (2.2), we conclude that B is 1-minor symmetric, therefore B is a K-matrix. □

Lemma 2.5 If $A=[a_{ij}]$ is an $n\times n$ $K_0$-matrix, then

$$\prod _{i=1}^n{a}_{ii}\ge {\left[\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}}\mathit{\text{for each }}\sigma \in S_n.$$

(2.3)

Proof For each $\sigma \in S_n$, by Lemma 2.3, we obtain

$$a_{i\sigma (i)}{a}_{\sigma (i)i}\ge 0\text{for }\mathrm{\forall }i=1,2,\dots ,n.$$

Since each $2\times 2$ principal minor of A is nonnegative, we have

$$a_{ii}{a}_{\sigma (i)i\sigma (i)}\ge a_{i\sigma (i)}{a}_{\sigma (i)i}\ge 0\text{for }\mathrm{\forall }i=1,2,\dots ,n,$$

thus

$${\left(\prod _{i=1}^n{a}_{ii}\right)}^2\ge \prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i},$$

hence inequality (2.3) follows. □

3 Main results

In this section, we give some new upper bounds for Fischer’s inequality and Hadamard’s inequality, and extend the corresponding results due to Zhang and Yang (see [11]).

Theorem 3.1 If $A=[a_{ij}]$ is an $n\times n$ $K_0$-matrix, with ${\prod }_{i=1}^n{a}_{ii}>0$, then

$$detA\le detA[\alpha ]\cdot detA\left[{\alpha }^c\right](1-\underset{j\in {\alpha }^c}{\underset{i\in \alpha }{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}}).$$

(3.1)

Proof If A is singular, (3.1) is valid obviously. If A is a nonsingular $K_0$-matrix, by Fisher’s inequality, we know that A is a P-matrix, therefore A is a K-matrix. Now we prove (3.1) for the K-matrix by induction on n. When $n=2$, it is very easy to see that the result is valid. We suppose that the result is valid for all $m\times m$ ($m<n$ and $n\ge 3$) K-matrices, let k be the cardinality of α. Note that α and ${\alpha }^c$ are symmetric in inequality (3.1). Without loss of generality, we assume $k\ge 2$. Let

$$\underset{j\in {\alpha }^c}{\underset{i\in \alpha }{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}}=\frac{a_{i_1{j}_1}{a}_{j_1{i}_1}}{a_{i_1{i}_1}{a}_{j_1{j}_1}},$$

where $i_1\in \alpha$ and $j_1\in {\alpha }^c$.

For any $r\in \alpha$ and $r\ne i_1$, let $\omega =\alpha -\{r\}$, and $S=(S_{ij})$ be the $(n-k+1)\times (n-k+1)$ Sylvester matrix of A associated with $A[\omega ]$. By Lemma 2.4, S is a K-matrix. Clearly, $S_{rr}=detA[\alpha ]$, then, by Fisher’s inequality, we have

$$detS\le S_{rr}\cdot detS\left[{\alpha }^c\right]=detA[\alpha ]\cdot detS\left[{\alpha }^c\right],$$

(3.2)

where ${\alpha }^c=〈n〉-\alpha$. It follows from Sylvester’s identity (2.1) and (3.2) that

$$detA=\frac{detS}{{(detA[\omega ])}^{n-k}}\le \frac{detA[\alpha ]\cdot detS[{\alpha }^c]}{{(detA[\omega ])}^{n-k}}.$$

(3.3)

Let $\beta =〈n〉-\{r\}$, from Sylvester’s identity (2.1), we obtain

$$detS\left[{\alpha }^c\right]={(detA[\omega ])}^{n-k-1}\cdot detA[\beta ].$$

(3.4)

Thus, by the inductive hypothesis, we have

$$detA[\beta ]\le detA[\omega ]\cdot detA\left[{\alpha }^c\right](1-\underset{j\in {\alpha }^c}{\underset{i\in \omega }{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}}).$$

(3.5)

From (3.3), (3.4) and (3.5), it follows that

$$detA\le detA[\alpha ]\cdot detA\left[{\alpha }^c\right](1-\underset{j\in {\alpha }^c}{\underset{i\in \omega }{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}}).$$

(3.6)

Clearly,

$$\underset{j\in {\alpha }^c}{\underset{i\in \omega }{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}}=\underset{j\in {\alpha }^c}{\underset{i\in \alpha }{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}}.$$

(3.7)

Combining (3.6) and (3.7), we obtain inequality (3.1). □

Example 3.1 Now, we consider a $4\times 4$ K-matrix that lies in none of the classes PD, TP, M and $M^{-1}$. Let

$$A=\left(\begin{array}{cccc}15& 5& 0& -3\\ 1& 3& 6& 9\\ 0& 3& 7& 11\\ -1& 5& 15& 30\end{array}\right).$$

Then A is 1-minor symmetric and A is a P-matrix. By Lemma 2.2, we know that A is a K-matrix. Let $\alpha =\{1,2\}$, then ${\alpha }^c=\{3,4\}$. By calculating, we have

$$detA=36<detA[\{1,2\}]\cdot detA[\{3,4\}](1-\underset{j\in \{3,4\}}{\underset{i\in \{1,2\}}{max}}\frac{a_{ij}{a}_{ji}}{a_{ii}{a}_{jj}})=\frac{1,800}{7},$$

therefore inequality (3.1) holds.

Let $A[\alpha ]=A[\{k\}]$ ($k\in 〈n〉$), by Theorem 3.1 and the induction, we can obtain the following conclusion.

Corollary 3.2 If $A=[a_{ij}]$ is an $n\times n$ $K_0$-matrix, with ${\prod }_{i=1}^n{a}_{ii}>0$, then

$$detA\le \underset{k\in 〈n〉}{min}{a}_{kk}\underset{i\ne k}{\overset{n}{\prod _{i=1}}}(a_{ii}-\frac{a_{ik}{a}_{ki}}{a_{kk}}).$$

If $A=[a_{ij}]$ is a totally nonnegative matrix with ${\prod }_{i=1}^n{a}_{ii}>0$, Corollary 3.2 is certainly valid, so Corollary 3.2 is the generalization of Theorem 3 in [11].

Theorem 3.3 If $A=[a_{ij}]$ is an $n\times n$ $K_0$-matrix, then

$$detA\le \prod _{i=1}^n{a}_{ii}-\underset{1\ne \sigma \in S_n}{max}{\left(\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right)}^{1/2}.$$

(3.8)

Proof If A is singular, by Lemma 2.3 and Lemma 2.5, we know (3.8) is valid. If A is a nonsingular $K_0$-matrix, then A is a K-matrix. We suppose that A is a K-matrix in the following.

If $n=2$, then equality in (3.8) holds. If $n>2$, then first we prove the following conclusion:

For any $1\ne \sigma \in S_n$, there exists $k_0\ne \sigma (k_0)$ such that

$${\left[\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}}\le a_{k_0\sigma (k_0)}{a}_{\sigma (k_0)k_0}\prod _{i=1,i\ne k_0,\sigma (k_0)}^n{a}_{ii}.$$

(3.9)

Case 1. If ${\prod }_{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}=0$. For any $1\ne \sigma \in S_n$, by Lemma 2.3, we know that (3.9) certainly holds.

Case 2. If ${\prod }_{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}>0$. Let $T=\{k|\sigma (k)\ne k\}$, then $|T|\ge 2$, it is easy to see that (3.9) is true even with the equality sign for $|T|=2$. Now we assume that $|T|>2$.

Suppose that there exists $1\ne \sigma \in S_n$, for any $k\ne \sigma (k)$, such that

$${\left[\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}}>a_{k\sigma (k)}{a}_{\sigma (k)k}\prod _{i=1,i\ne k,\sigma (k)}^n{a}_{ii}.$$

(3.10)

Then multiplying all the possible inequalities in (3.10) yields

$${\left[\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{|T|}{2}}>\prod _{k\in T}\left[a_{k\sigma (k)}{a}_{\sigma (k)k}\prod _{i=1,i\ne k,\sigma (k)}^n{a}_{ii}\right].$$

(3.11)

Let $S=〈n〉-T$. If $S\ne \mathrm{\varnothing }$, by (3.11) we have

$${\left[\prod _{i\in T}{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{|T|}{2}}{\left[\prod _{i\in S}{a}_{ii}\right]}^{|T|}>\left[\prod _{i\in T}{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]\cdot {\left[\prod _{i\in S}{a}_{ii}\right]}^{|T|}\prod _{i\in T}{a}_{ii}^{|T|-2}.$$

(3.12)

If $S=\mathrm{\varnothing }$, by (3.11) we have

$${\left[\prod _{i\in T}{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{|T|}{2}}>\left[\prod _{i\in T}{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]\cdot \prod _{i\in T}{a}_{ii}^{|T|-2}.$$

(3.13)

By (3.12) and (3.13), we have

$${\left[\prod _{i\in T}{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}}>\prod _{i\in T}{a}_{ii}.$$

(3.14)

On the other hand, since the principal submatrix $A[T]$ of A is a K-matrix, applying Lemma 2.5 to $A[T]$ yields

$${\left[\prod _{i\in T}{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}}\le \prod _{i\in T}{a}_{ii},$$

(3.15)

which is a contradiction to (3.14), therefore (3.9) holds.

By Fisher’s inequality (3.1) and (3.9), we have

$$\begin{array}{rl}detA& \le detA\left[\{k_0,\sigma (k_0)\}\right]\cdot detA\left[{\{k_0,\sigma (k_0)\}}^c\right]\\ \le [a_{k_0{k}_0}{a}_{\sigma (k_0)\sigma (k_0)}-a_{\sigma (k_0)k_0}{a}_{k_0\sigma (k_0)}]\cdot \prod _{i=1,i\ne k_0,\sigma (k_0)}{a}_{ii}\\ \le \prod _{i=1}^n{a}_{ii}-{\left[\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}},\end{array}$$

therefore (3.8) holds. □

If $A=[a_{ij}]$ is a totally nonnegative matrix, Theorem 3.3 is certainly valid too, so Theorem 3.3 is the generalization of Theorem 4 in [11].

Example 3.2 Now we consider the previous K-matrix. Let

$$A=\left(\begin{array}{ccc}4& 2& -3\\ 1& 3& 1\\ -1& 1& 2\end{array}\right).$$

By calculating we have

$$detA=2\le \prod _{i=1}^3{a}_{ii}-\underset{1\ne \sigma \in S_3}{max}{\left[\prod _{i=1}^3{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right]}^{\frac{1}{2}}=24-max\{4,9,\sqrt{6},4,\sqrt{6}\}=15,$$

therefore inequality (3.8) holds.

By Corollary 3.2 and Theorem 3.3, we get the following result.

Corollary 3.4 If $A=[a_{ij}]$ is an $n\times n$ $K_0$-matrix, with ${\prod }_{i=1}^n{a}_{ii}>0$, then

$$detA\le min\{\prod _{i=1}^n{a}_{ii}-\underset{1\ne \sigma \in S_n}{max}{\left(\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right)}^{1/2},\underset{k\in 〈n〉}{min}{a}_{kk}\underset{i\ne k}{\overset{n}{\prod _{i=1}}}(a_{ii}-\frac{a_{ik}{a}_{ki}}{a_{kk}})\}.$$

If A is a totally nonnegative matrix, Corollary 3.4 is valid, so we obtain the result in [11].

Corollary 3.5 [11]

If $A=[a_{ij}]$ is an $n\times n$ totally nonnegative matrix, with ${\prod }_{i=1}^n{a}_{ii}>0$, then

$$detA\le min\{\prod _{i=1}^n{a}_{ii}-\underset{1\ne \sigma \in S_n}{max}{\left(\prod _{i=1}^n{a}_{i\sigma (i)}{a}_{\sigma (i)i}\right)}^{1/2},\underset{k\in 〈n〉}{min}{a}_{kk}\underset{i\ne k}{\overset{n}{\prod _{i=1}}}(a_{ii}-\frac{a_{ik}{a}_{ki}}{a_{kk}})\}.$$