We know $$1 = \lVert A \cdot A^{-1} \rVert \leq \lVert A \rVert \cdot \lVert A^{-1} \rVert$$ do we have the upper bound for $\lVert A\rVert\cdot\lVert A^{-1}\rVert$ ?
2026-04-25 01:00:29.1777078829
upper bound of the product of a matrix norm and its inverse norm
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Yes: it's $1$ for $1\times1$ matrices and $+\infty$ for $m\times m$ matrices with $m\ge2$: in fact, consider the sequence $$A_n=\begin{pmatrix}n&0&0\\ 0&\frac1n&0\\ 0&0&I_{m-2}\end{pmatrix}$$