Norm inequalities for product of matrices

IF 0.6 Q3 MATHEMATICS ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-03-14 DOI:10.1007/s44146-024-00121-1

Ahmad Al-Natoor

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Abstract

In this paper, we prove some new norm inequalities for product of matrices. Among other results, we prove that if A and B are n $\times $ n complex matrices, then

$$\begin{aligned} \left| \left| \left| \text { }\left| AB^{*}\right| ^{2}\right| \right| \right| \le \min (\left| \left| \left| B^{*}B\right| \right| \right| \left\| A^{*}A\right\| ,\left| \left| \left| A^{*}A\right| \right| \right| \left\| B^{*}B\right\| ). \end{aligned}$$

$$\begin{aligned} \left\| AB^{*}\right\| ^{2}\le \left\| A^{*}A\right\| \left\| B^{*}B\right\| , \end{aligned}$$

which is known as the Cauchy–Schwarz inequality. Also, we prove that if A and B are n $\times $ n complex matrices, then

$$\begin{aligned} \text { }\left\| AB^{*}\right\| ^{2}\le w\left( A^{*}AB^{*}B\right) , \end{aligned}$$

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矩阵积的规范不等式

本文证明了矩阵乘积的一些新的范数不等式。在其他结果中，我们证明了如果A和B是n $\times $ n个复矩阵，则$$\begin{aligned} \left| \left| \left| \text { }\left| AB^{*}\right| ^{2}\right| \right| \right| \le \min (\left| \left| \left| B^{*}B\right| \right| \right| \left\| A^{*}A\right\| ,\left| \left| \left| A^{*}A\right| \right| \right| \left\| B^{*}B\right\| ). \end{aligned}$$，特别是如果$\left| \left| \left| \cdot \right| \right| \right| =\left\| \cdot \right\| ,$则$$\begin{aligned} \left\| AB^{*}\right\| ^{2}\le \left\| A^{*}A\right\| \left\| B^{*}B\right\| , \end{aligned}$$，这被称为Cauchy-Schwarz不等式。同时，我们证明了如果A和B是n $\times $ n个复矩阵，则$$\begin{aligned} \text { }\left\| AB^{*}\right\| ^{2}\le w\left( A^{*}AB^{*}B\right) , \end{aligned}$$是上述Cauchy-Schwarz不等式的一种细化。这里$ \left| \left| \left| \cdot \right| \right| \right| ,$$\left\| \cdot \right\| ,$和$w(\cdot )$分别表示矩阵的任意酉不变范数、谱范数和数值半径。

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