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Let $s_{1},s_{2} : s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, where $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge0$s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that $$\left|s_{1}\left(M+N\right)-s_{1}\left(N\right)\right|+\left|s_{2}\left(M+N\right)-s_{2}\left(N\right)\right|\leq s_{1}\left(M\right)+s_{2}\left(M\right)$$ for any two $2\times2$ real matrices $M$ and $N$?

Thanks in advance for any helpful answers.
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A sigular singular value inequality

Let $s_{1},s_{2} : \mathbb{R}^{2\times 2} \mapsto \mathbb{R}+$, mathbb{R}_+$, where $s{1}\left(\cdot\right)\ge s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge0$, be the singular values of a $2\times2$ matrix. Is it true that $$\left|s_{1}\left(M+N\right)-s_{1}\left(N\right)\right|+\left|s_{2}\left(M+N\right)-s_{2}\left(N\right)\right|\leq s_{1}\left(M\right)+s_{2}\left(M\right)$$ for any two $2\times2$ real matrices $M$ and $N$?

Thanks in advance for any helpful answers.
show/hide this revision's text 1

A sigular value inequality

Let $s_{1},s_{2} : \mathbb{R}^{2\times 2} \mapsto \mathbb{R}+$, $s{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge0$, be the singular values of a $2\times2$ matrix. Is it true that $$\left|s_{1}\left(M+N\right)-s_{1}\left(N\right)\right|+\left|s_{2}\left(M+N\right)-s_{2}\left(N\right)\right|\leq s_{1}\left(M\right)+s_{2}\left(M\right)$$ for any two $2\times2$ real matrices $M$ and $N$?

Thanks in advance for any helpful answers.