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Let A and B be two hermitian matrices. Let $|||\cdot|||$ be any induced matrix norm. I hope to find some upper bound inequalities or relationship of the matrix norm: $|||iA - B|||$.

The only thing I know is that it is $|||iA - B|||\leq |||A||| + |||B|||$.

Which of the following is also true?

1) $|||iA - B||| \leq max(|||A|||, |||B|||)$

2) $|||iA - B||| \leq min(|||A|||, |||B|||)$

3) $|||iA - B||| \leq abs(|||A||| - |||B|||)$

Any other interesting upper bounds or relationship?

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  • $\begingroup$ $iA-B$ refers to the matrix with entries of the form $-b_{ij} + i \cdot a_{ij}$, right? $\endgroup$ – Laray Feb 10 '17 at 9:45
  • $\begingroup$ @Laray yes the imaginary number multiplies all entries of A. $\endgroup$ – Ka-Wa Yip Feb 10 '17 at 9:59
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None of the three inequalities you’ve shown hold in general for Hermitian matrices $A$ and $B$.

To demonstrate this, we can look at the case where $A$ and $B$ are $1\times 1$ real matrices (trivially Hermitian), in which case the induced matrix norm reduces to an absolute value. Writing $A=a$ and $B=b$ for $a,b\in \mathbb{R}$, we have $|||iA - B|||=\sqrt {a^2 + b^2 }$. Then, for these $A,B\in \mathbb{R^{1\times 1}}$, we have that

$$ \mathop{\min } (|||A|||, |||B|||)\leq \mathop {\max } (|||A|||, |||B|||)\leq |||iA - B|||$$

and

$$ abs(|||A||| - |||B|||)\leq |||iA - B|||,$$

where the rightmost inequalities are strict for $ab \ne 0.$

We can also use this special case to generate counterexamples to the proposed inequalities for $A,B\in\mathbb{R^{n\times n}}$ with $n>1$ by (for example) setting $a_{11}=a$ and $b_{11}=b$ for $A$ and $B$ with all the other matrix entries set to $0.$

Without further restrictions on $A$ and $B$, you’re unlikely to find a better general upper bound than $|||iA - B|||\leq |||A||| + |||B|||$.

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