Determinantal inequalities for block Hadamard product and Khatri-Rao product of positive definite matrices

1.   Introduction

Let $\mathbb{M}_{n}$ be the set of $n\times n$ complex matrices. If $X$ is positive semidefinite, we put $X\geq 0$. For two Hermitian matrices $X, Y\in\mathbb{M}_{n}, X\geq Y$ means $X-Y$ is positive semidefinite. If $X$ is positive definite, we put $X > 0$.

The Hadamard product of $A, B\in \mathbb{M}_{n}$ is denoted by $A\circ B$, and the Hadamard product of $A_{1}, \ldots, A_{m}\in \mathbb{M}_{n}$ is denoted by $\prod^{m}_{i = 1}\circ A_{i}$. The Kronecker product of $A$ and $B$ is denoted by $A\otimes B = (a_{ij}B)$. If $A = \left(\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array} \right)\in \mathbb{M}_{n}$ with $A_{11}$ nonsingular, then the Schur complement of $A_{11}$ in $A$ is defined as $A/A_{11} = A_{22}-A_{21}A_{11}^{-1}A_{12}.$ A well known property of the Schur complement is

The set of all complex matrices partitioned as $p\times p$ blocks with each block $q\times q$ is denoted by $\mathbb{M}_{p}(\mathbb{M}_{q})$. Let $\textbf{A} = (A_{ij}), \textbf{B} = (B_{ij})\in \mathbb{M}_{p}(\mathbb{M}_{q})$. The block Hadamard product of $\textbf{A}$ and $\textbf{B}$ is given by $\textbf{A}\Box \textbf{B}: = (A_{ij}B_{ij})$, where $A_{ij}B_{ij}$ denotes the usual matrix product of $A_{ij}$ and $B_{ij}$. If every block of $\textbf{A}$ commutes with corresponding block of $\textbf{B}$, we say that $\textbf{A}, \textbf{B}$ block commute. The Khatri-Rao product of $\textbf{A}$ and $\textbf{B}$ is given by $\textbf{A}\ast \textbf{B}: = (A_{ij}\otimes B_{ij})$. We denote by $\textbf{I}_{\textbf{p}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$ the $pq\times pq$ identity matrix, which is partitioned according to the block structure of the matrices of $\mathbb{M}_{p}(\mathbb{M}_{q})$. Clearly, when $q = 1$, that is, A and B are $p\times p$ matrices with complex entries, the block Hadamard product and the Khatri-Rao product coincide with the Hadamard product. These matrix products have been used in many fields, such as matrix analysis, statistical analysis, communication and information theory, etc. For more information, we refer to ^{[3,4,5,6,9,10]}.

Let $A_{i}\in \mathbb{M}_{n}, i = 1, \ldots, m,$ be positive definite matrices whose diagonal blocks are $n_{j}$-square matrices $A^{(j)}_{i}, j = 1, \ldots, k$ (so $n_{1}+\cdots+n_{k} = n$). Denote $\sum_{i = 1}^{m}$ or $\prod_{i = 1}^{m}\circ$ by $\star$, then

follows directly from Fischer's inequality [2,p. 506].

By making use of a result of Lin ^[7], Choi ^[1] proved the following inquality,

Later, Liu et al. ^[8] gave a new proof of Choi's inequality(1.2), and they also obtained the following theorem.

Theorem 1. Let $A_{i}\in \mathbb{M}_{n}, i = 1, \ldots, m,$ be positive definite whose diagonal blocks are $n_{j}$-square matrices $A^{(j)}_{i}$ for $j = 1, \ldots, k$ (so $n_{1}+\cdots+n_{k} = n$). Then

In this paper, we give an alternative proof of Theorem 1, this is done in Section 2. In Section 3, we present the following inequality for the block Hadamard product. Clearly, when $q = 1$, Theorem 2 reduces to Theorem 1.

Theorem 2. Let ${\textbf{A}}_{i}\in \mathbb{M}_{p}(\mathbb{M}_{q})$, $i = 1, \ldots, m$, partition ${\textbf{A}}_{i}$ with diagonal blocks ${\textbf{A}}^{(j)}_{i}\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$ (so $p_{1}+\cdots+p_{t} = p$). If ${\textbf{A}}_{i}, i = 1, \ldots, m,$ are positive definite and block commute, then

The result for the Khatri-Rao product is given in Section 4.

2.   Alternative proof of Theorem 1

We list some lemmas which are important for our proof.

Lemma 1. [2,Corollary 7.7.4] If $A, B \in \mathbb{M}_{n}$ such that $A\geq B > 0$, then $\det A\geq \det B$.

Lemma 2. [2,Theorem 7.5.3] Let $A, B\in \mathbb{M}_{n}$ be positive semidefinite. Then $A\circ B\geq 0$.

Lemma 3. [11,Theorem 7.13] Let $A\in \mathbb{M}_{n}$ be positive definite. Partition $A$ as $A = \left(\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array} \right)$ with $A_{11}$ square. Let $A^{-1}$ be conformally partitioned as $A$, then

$(1)\; (A_{ii})^{-1}\leq (A^{-1})_{ii}, i = 1, 2;$

$(2)\; A^{-1}/(A^{-1})_{11} = (A_{22})^{-1}$.

Lemma 4. [2,p.504] Let $A, B\in \mathbb{M}_{n}$ be positive definite. Partition $A, B$ as $A = \left(\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array} \right)$, $B = \left(\begin{array}{cc} B_{11} & B_{12} \\ B_{21} & B_{22} \\ \end{array} \right)$ with $A_{11}, B_{11}\in \mathbb{M}_{k}$, $1\leq k < n$, then

By Lemma 4 and induction, we have

Lemma 5. Let $A_{i}\in\mathbb{M}_{n} , i = 1, \ldots, m$, be positive definite and conformally partitioned, and $A^{(1)}_{i}$ be the $(1, 1)$ block of $A_{i}$. Then

Now we are ready to present.

Proof of Theorem 1. For all $i = 1, \ldots, m,$ as $A_{i}$ are positive definite, $A^{-1}_{i}$ are all positive definite. Partition $A^{-1}_{i}$ as $A_{i}$ for $i = 1, \ldots, m,$ the diagonal blocks of $A^{-1}_{i}$ are $n_{j}$-square matrices $(A^{-1}_{i})^{(j)}$ for $j = 1, \ldots, p$. Using mathematical induction on $k$, we may assume $k = 2$. By Lemma 5 and Lemma 1, we get

Then we have

where the first equality above is by (1.1); the first inequality is by (2.1); the second inequality is by Lemma 3(1); the last equality is due to Lemma 3(2).

3.   Proof of Theorem 2

In order to prove Theorem 2, we need to show the following lemmas.

Lemma 6. [4,Corollary 3.3] Let ${\textbf{A}}, {\textbf{B}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$. If ${\textbf{A}}, {\textbf{B}}$ are positive semidefinite andblock commute, then ${\textbf{A}}\Box {\textbf{B}}\geq 0$.

Lemma 7. [4,Lemma 2.4] Let ${\textbf{A}}, {\textbf{B}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$. If ${\textbf{B}}$ is invertible, then ${\textbf{A}}, {\textbf{B}}$ block commute if and only if ${\textbf{A}}, {\textbf{B}}^{-1}$ block commute.

Lemma 8. Let ${{\textbf{A}}} = \left(\begin{array}{cc} {\textbf{A}}_{11} & {\textbf{A}}_{12} \\ {\textbf{A}}_{12}^\ast & \textbf{A}_{22} \\ \end{array} \right), {{\textbf{B}}} = \left(\begin{array}{cc} {\textbf{B}}_{11} & {\textbf{B}}_{12} \\ {\textbf{B}}_{12}^\ast & {\textbf{B}}_{22} \\ \end{array} \right)\in \mathbb{M}_{p}(\mathbb{M}_{q})$ with ${\textbf{A}}_{11}, {\textbf{B}}_{11}\in \mathbb{M}_{h}(\mathbb{M}_{q}), h < p$. If ${\textbf{A}}, {\textbf{B}}$ are positive definite and block commute, then

Proof. Let

and

Then $\textbf{E}$ and $\textbf{F}$ are positive semidefinite and block commute.

By Lemma 6, we get that

is positive semidefinite, thus

that is

which implies

It follows from

that

This completes the proof.

Lemma 9. Let ${{\textbf{A}}}_{i} = \left(\begin{array}{cc} ({\textbf{A}}_{i})_{11} & ({\textbf{A}}_{i})_{12} \\ ({\textbf{A}}_{i})_{12}^\ast & ({\textbf{A}}_{i})_{22} \\ \end{array} \right)\in \mathbb{M}_{p}(\mathbb{M}_{q})$ with $({\textbf{A}}_{i})_{11}\in \mathbb{M}_{h}(\mathbb{M}_{q}), h < p, i = 1, \ldots, m$. If ${\textbf{A}}_{i}, i = 1, \ldots, m,$ are positive definite and block commute, then

Proof. By Lemma 8 and induction, we can get

Then

Now we give the proof of Theorem 2.

Proof of Theorem 2. For all $i = 1, \ldots, m,$ as $\textbf{A}_{i}$ are positive definite, $\textbf{A}^{-1}_{i}$ are all positive definite. Partition $\textbf{A}^{-1}_{i}$ as $\textbf{A}_{i}$ for $i = 1, \ldots, m,$ then the diagonal blocks of $(\textbf{A}^{-1}_{i})^{(j)}\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$ for $j = 1, \ldots, t$. By Lemma 7, we get that $\textbf{A}^{-1}_{i}$, $i = 1, \ldots, m,$ block commute. Using mathematical induction on $t$, we may assume $t = 2$. By Lemma 9, we get

By Lemma 3, we have

This completes the proof.

Corollary 1. Let ${\textbf{A}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$ be positive definite, partition ${\textbf{A}}$ with diagonal blocks ${\textbf{A}}^{(j)}$$\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$ (so $p_{1}+\cdots+p_{t} = p$). Then

Corollary 2. Let ${\textbf{A}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$ be positive definite, partition ${\textbf{A}}$ with diagonal blocks ${\textbf{A}}^{(j)}$$\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$ (so $p_{1}+\cdots+p_{t} = p$). Let ${\textbf{B}}^{(j)}$$\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$, be positive semidefinite, ${\textbf{B}} = diag({\textbf{B}}^{(1)}, \ldots, {\textbf{B}}^{(t)})$, ${\textbf{A}}$ and ${\textbf{B}}$ block commute. Then

Proof. Assume that $\textbf{B}$ is nonsingular, that is, $\textbf{B}^{(j)}$ are all invertible for $j = 1, \ldots, t$. Then, (3.1) follows by Lemma 7 and Theorem 2. By a standard continuity argument, the statement is also true if $\textbf{B}$ is singular.

When $t = p$ in Theorem 2, we get the following.

Corollary 3. Let ${{\textbf{A}}}_{i}\in \mathbb{M}_{p}(\mathbb{M}_{q}), i = 1, \ldots, m$, with diagonal blocks $q$-square matrices $A^{(j)}_{i}$, $j = 1, \ldots, p$. If ${\textbf{A}}_{i}, i = 1, \ldots, m,$ are positive definite and block commute, then

4.   Results for the Khatri-Rao product

The following lemmas will be used in the main result of this section.

Lemma 10. [9,Theorem 5] Let ${\textbf{A}}\geq{\textbf{B}}\geq 0, {\textbf{C}}\geq{\textbf{D}}\geq0$, and ${\textbf{A}}, {\textbf{B}}, {\textbf{C}}$ and ${\textbf{D}}$ be compatibly partitioned matrices. Then

Lemma 11. [5,Lemma 3.3] Let ${{\textbf{A}}} = \left(\begin{array}{cc} {\textbf{A}}_{11} & {\textbf{A}}_{12} \\ {\textbf{A}}_{12}^\ast & {\textbf{A}}_{22} \\ \end{array} \right), {{\textbf{B}}} = \left(\begin{array}{cc} {\textbf{B}}_{11} & {\textbf{B}}_{12} \\ {\textbf{B}}_{12}^\ast & {\textbf{B}}_{22} \\ \end{array} \right)\in \mathbb{M}_{p}(\mathbb{M}_{q})$ with ${\textbf{A}}_{11}, {\textbf{B}}_{11}\in \mathbb{M}_{h}(\mathbb{M}_{q}), h < p$. If ${\textbf{A}}, {\textbf{B}}$ are positive definite, then

By Lemma 11 and induction, we have

Lemma 12. Let ${{\textbf{A}}}_{i} = \left(\begin{array}{cc} ({\textbf{A}}_{i})_{11} & ({\textbf{A}}_{i})_{12} \\ ({\textbf{A}}_{i})_{12}^\ast & ({\textbf{A}}_{i})_{22} \\ \end{array} \right)\in \mathbb{M}_{p}(\mathbb{M}_{q})$ with $({\textbf{A}}_{i})_{11}\in \mathbb{M}_{h}(\mathbb{M}_{q}), h < p, i = 1, \ldots, m$. If ${\textbf{A}}_{i}, i = 1, \ldots, m,$ are positive definite, then

Now, we give the result for the Khatri-Rao product. Clearly, when $q = 1$, Theorem 3 reduces to Theorem 1.

Theorem 3. Let ${{\textbf{A}}}_{i}\in \mathbb{M}_{p} \; (\mathbb{M}_{q})$, $i = 1, \ldots, m$, partition ${\textbf{A}}_{i}$ with diagonal blocks ${\textbf{A}}^{(j)}_{i} \; \in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$ (so $p_{1}+\cdots+p_{t} = p$). If ${\textbf{A}}_{i}$, $i = 1, \ldots, m,$ are positive definite, then

Proof. This follows analogous steps to the proof of Theorem 2.

Corollary 4. Let ${\textbf{A}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$ be positive definite, partition ${\textbf{A}}$ with diagonal blocks ${\textbf{A}}^{(j)}$$\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$ (so $p_{1}+\cdots+p_{t} = p$). Then

and

Corollary 5. Let ${\textbf{A}}\in \mathbb{M}_{p}(\mathbb{M}_{q})$ be positive definite, partition ${\textbf{A}}$ with diagonal blocks ${\textbf{A}}^{(j)}$$\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$ (so $p_{1}+\cdots+p_{t} = p$). Let ${\textbf{B}}^{(j)}$$\in \mathbb{M}_{p_{j}}(\mathbb{M}_{q})$, $j = 1, \ldots, t$, be positive semidefinite, ${\textbf{B}} = diag({\textbf{B}}^{(1)}, \ldots, {\textbf{B}}^{(t)})$. Then

and

When $t = p$ in Theorem 3, we get the following.

Corollary 6. Let ${{\textbf{A}}}_{i}\in \mathbb{M}_{p}(\mathbb{M}_{q}), i = 1, \ldots, m$, with diagonal blocks $q$-square matrices $A^{(j)}_{i}$, $j = 1, \ldots, p$. If ${\textbf{A}}_{i}, i = 1, \ldots, m,$ are positive definite, then

Acknowledgments

The work was supported by National Natural Science Foundation of China (NNSFC) [Grant Number 11971294].

Conflict of interest

We declare no conflict of interest.