Some matrix inequalities related to norm and singular values

1.   Introduction

Let $M_{n}$ be the space of $n \times n$ complex matrices. If $A$ is a Hermitian element of $M_{n}$, then we enumerate its eigenvalues as $\lambda_{1}(A) \geq \cdots \geq \lambda_{n}(A)$ (see ^[1] for more details). The singular values of $A \in M_{n}$ are defined to be the nonnegative square roots of the eigenvalues of $A^{\ast} A$, where $A^{\ast}$ denotes the conjugate transpose of a matrix $A$, i.e., $s_{i}(A) = \lambda_{i}(|A|)$ ($1 \leq i \leq n$) for $|A| = \left(A^{\ast} A\right)^{\frac{1}{2}}$. The notation $A\geq (>)$ $0$ is used to mean that $A$ is positive semi-definite (positive definite).

Given a real vector $x = \left(x_{1}, x_{2}, \cdots, x_{n}\right) \in R^{n}$, we rearrange its components as $x_{[1]}\geq x_{[2]}\geq \cdots \geq x_{[n]}$. For $x = \left(x_{1}, x_{2}, \cdots, x_{n}\right)$, $y = \left(y_{1}, y_{2}, \cdots, y_{n}\right) \in R^{n}$, we say $x$ is weakly majorized by $y$ $\left(x\prec _{w}y\right)$ if $\sum_{i = 1}^{k}x_{[i]}\leq \sum_{i = 1}^{k}y_{[i]}$ for $k = 1, 2, \cdots, n$. If $x\prec _{w}y$ and $\sum_{i = 1}^{n}x_{i} = \sum_{i = 1}^{n}y_{i}$ hold, we say $x$ is majorized by $y$ and denote $x\prec y$.

For $x = \left(x_{1}, x_{2}, \cdots, x_{n}\right)$ with $x_{i} \geq 0$ for $1\leq i \leq n$, we write $x\in R_{+}^{n}$. Let $x, y\in R_{+}^{n}$. The weak log-majorization $x\prec _{w\log }y$ can be defined as

The log-majorization $x\prec _{\log }y$ holds if, and only if, $x\prec _{w\log }y$ and $\prod_{i = 1}^{n}x_{i} = \prod_{i = 1}^{n}y_{i}.$

Let $A, B \geq 0$. The weak log-majorization inequality

for any complex number $z$ was proved by Zhan ^[2].

Next, in ^[3], the inequality (1.1) was extended to the form

where $A_{i}$ are normal matrices, $i = 1, 2, \cdots, m.$

For $t\in \lbrack 0, 1]$, the $t$-geometric mean of $A, B\in M_{n}$ with $A, B$ are positive definite and defined as $A\#_{t}B = A^{\frac{1}{2}}\left(A^{-\frac{1}{2} }BA^{-\frac{1}{2}}\right) ^{t}A^{\frac{1}{2}}$ (^[4]). Their geometric mean is $A\#B = A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right) ^{\frac{1}{2}}A^{\frac{1}{2}}$ and a matrix Cauchy-Schwarz inequality for positive definite matrices $A_{i}$ and $B_{i}$ $(i = 1, 2, \cdots, n)$ is

also, see ^[5].

A norm $\|\cdot\|$ on $M_{n}$ is called unitarily invariant if $\|U A V\| = \|A\|$ for any $A \in M_{n}$ and any unitary $U, V \in M_{n}$. Fan's dominance principle ^[5] illustrates the relevance of majorization in matrix theory: For $A, B$ in $M_{n}$, the weak majorization $s(A) \prec_{w} s(B)$ means $\|A\| \leq\|B\|$ for all unitarily invariant norms $\|\cdot\|$ (see ^[4] for more details).

Norm inequality for sums of positive semi-definite matrices shown by M. Hayajneh, S. Hayajneh, and F. Kittaneh ^[6] can be stated as follows:

where $A_{i}, B_{i} \in M_{n}$ $(i = 1, 2, \cdots, m)$ are positive semi-definite matrices and $A_{i}$ commutes $B_{i}$ for each $i$. Inequality (1.4) is a refinement of the following inequality obtained by Audenaert ^[7]:

Zhao and Jiang ^[8] derived a generalization of inequality (1.4),

where $r \geq 1$.

Let $A, B\in M_{n}$ be positive semi-definite and suppose that $\frac{1}{p}+ \frac{1}{q} = 1$, $p$, $q > 1$, $a\in (0, 1)$. Wu proved in ^[9] that if $r\geq \max \left\{ \frac{1}{p}, \frac{1}{q}\right\}$, then

It is natural to raise the question that if $r < \max \left\{ \frac{1}{p}, \frac{1}{q}\right\}$, then does inequality (1.7) hold or not?

Zhang ^[3] utilized the compound matrix technique to derive inequality (1.2). In this paper, we present a new proof of inequality (1.2). We also give a generalization of inequality (1.6). Finally, we present some numerical examples to show that inequality (1.7) is not always true when $r < \max \left\{ \frac{1}{p}, \frac{1}{q}\right\}$.

2.   Main results

We begin this section with the following lemmas, which play an important role in our discussion.

Lemma 1. ^[5] Let $A\in M_{n}$, then

where the maximum is taken over all $n\times k$ matrices $W$ such that $W^{\ast}W = I$.

Lemma 2. ^[8] Let $\left(\begin{array}{cc} A & X \\ X^{\ast } & B \end{array} \right) \geq 0$, then

Lemma 3. ^[10] Let $p > 0$, $t \in[0, 1]$, then

We give a new proof of inequality (1.2).

Theorem 4. Let $A_{i}\in M_{n}$ be normal matrices $A_{i}$ ($i = 1, 2, \cdots, m$), then

Proof. An application of the polar decomposition reveals $\left(\begin{array}{cc} \left\vert A_{i}^{\ast }\right\vert & A_{i} \\ A_{i}^{\ast } & \left\vert A_{i}\right\vert \end{array} \right) \geq 0$ for any $i$. Hence,

is positive semi-definite. It follows from $\left\vert A_{i}^{\ast }\right\vert = \left\vert A_{i}\right\vert$ that $\left(\begin{array}{cc} \sum_{i = 1}^{m}\left\vert A_{i}\right\vert & \sum_{i = 1}^{m}A_{i} \\ \sum_{i = 1}^{m}A_{i}^{\ast } & \sum_{i = 1}^{m}\left\vert A_{i}\right\vert \end{array} \right) \geq 0$.

For all $n\times k$ matrices $W$ with $W^{\ast} W = I$,

Using Lemmas 1 and 2, we obtain

□ Next, we give a generalization of inequality (1.6).

Theorem 5. Let $A_{i}$, $B_{i} \in M_{n}$ be positive semi-definite matrices, then

for $r > 0$.

Proof. We first consider the case $A_{i}, B_{i} > 0$ $(i = 1, 2, \cdots, m).$ Using inequality (1.3), we get

for $k = 1, 2, \cdots, n.$

It follows from Lemma 3 that

for $k = 1, 2, \cdots, n.$

For the general case, by replacing $A_{i}$ and $B_{i}$ by $\varepsilon I_{n}+A_{i}$ and $\varepsilon I_{n}+B_{i}$ ($\varepsilon > 0$) for $i = 1, 2, \cdots, m,$ respectively, and repeating the same process as above, we obtain that

By continuity, we get the desired inequality. □

Finally, we show that

isn't always true if $r < \max \left\{ \frac{1}{p}, \frac{1}{q}\right\}$.

Using $\lambda _{j}\left(AB\right) \leq \lambda _{j}\left(\frac{A^{\frac{1 }{2}}+B^{\frac{1}{2}}}{2}\right) ^{4}$ (see ^[11]), we obtain

for $r = \frac{1-2\varepsilon }{2}$ $\left(0 < \varepsilon \leq \frac{1}{2}\right)$.

Inequality (1.5) is equivalent to

Inequality (2.4) can be rewritten as

for $1\leq k\leq n$. By Ky Fan's dominance principce ^[5], we see inequality (2.5) is equivalent to

Inequality (2.6) implies inequality (1.7) is true if $p = q = 2$, $a = \frac{1}{2}$, and $r < \max \left\{ \frac{1}{p}, \frac{1}{q}\right\}.$

Example 6. Let $A = \left[ \begin{array}{cc} 1 & 2 \\ 2 & 7 \end{array} \right], B = \left[ \begin{array}{cc} 1 & -1 \\ -1 & 4 \end{array} \right]$, $a = 0.12$, and $r = 0.22$ in inequality (1.7).

By calculating, we obtain $s_{1}^{r}\left(BA^{2}B\right) +s_{2}^{r}\left(BA^{2}B\right) \approx 4.9044$ and

Therefore, inequality (1.7) isn't true in this case.

3.   Conclusions

Matrix inequalities play important roles in linear algebra and it is of interest to study the properties of Positive semi-definite matrix. In this paper, we have presented a norm inequalities related to normal matrices by using block matrix technique. Next, a weak majorization inequality for $t$-geometric mean was established. Lastly, a numerical example has been provided to illustrate the necessity of a condition in an existing inequality.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no competing interests.