# Lower bound on matrix norm with a non-zero constant entry

Consider a real matrix $$A \in \mathbb{R}^{n \times n}$$, and its $$(i,j)$$ entry is a non-zero constant, denoted by $$a^0_{ij}$$. The other entries of $$A$$ are variables taken any real value.

I am wondering if there exists a positive constant $$r$$, such that for any $$A$$, $$||A||\geqslant r.$$ I am not specifying the norm here as I am currently looking for a general result that holds for any norm. (But the spectral norm is my primary interest)

An intuitive guess for the lower bound would be as follows. Let $$\bar{A}$$ be the matrix with $$(i,j)$$ entry as $$a^0_{ij}$$ and the other other entries as zero. Does the following relationship hold for any matrix norm?: For any $$A$$, $$||A|| \geqslant ||\bar{A}||.$$

I think the above inequality is clear if we consider an entry-wise norm. However, I am not sure if it holds for an operator norm.

• There is no such lower bound for any norm. To build counterexamples, note that for any invertible $T$ and norm $\|\cdot\|$, the function $\|A\|_T = \|TAT^{-1}\|$ defines a new norm. Commented May 3, 2020 at 17:31
• Thank you for your comment, but I do not understand why that is a counterexample. By the way, Is it possible to find the constant $r$ for the spectral norm? Commented May 3, 2020 at 18:17
• I did not give you a counterexample, I have just told you that a counterexample can be obtained in this way. I suspect that a random choice has a high probability of producing a counterexample Commented May 3, 2020 at 18:19
• I suspect that there is such a constant for the spectral norm Commented May 3, 2020 at 18:20
• I found $r$ as a function of $a^0_{ij}$ for $2 \times 2$ matrices, but have not found a general statement for $n \times n$ matrices. Commented May 3, 2020 at 18:24

As a consequence of the interlacing theorem for singular values (see also Bhatia's Matrix Analysis), it turns out that $$\sigma_{\max}(A) \geq \sigma_{\max}(A')$$ whenever $$A'$$ is a submatrix of $$A$$.
It follows that the spectral norm satisfies $$\|A\| \geq |a_{ij}|$$ for any entry $$a_{ij}$$ of $$A$$.