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.
Best Answer
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 $A$.
I suspect that a similar statement can be made for arbitrary orthogonally/unitarily invariant norms.