For $n>2$, are there norms $\parallel.\parallel\_{a}$ and $\parallel.\parallel\_{b}$ on $M\_{n}(\mathbb{R})$ with the following property:



> 
> $A\in M\_{n}(\mathbb{R})$ is singular if and only if $\parallel A \parallel\_{a}=\parallel A \parallel\_{b}$ 
> 
> 
> 


For $n=2,\;$ these norms are $\parallel A \parallel\_{a}=\sqrt{\sum a\_{ij}^{2}}$ and
$\parallel A \parallel\_{b}=\parallel.\parallel\_{op}$, the operator norm.



A norm description for singular matrices
(This answer expands the ideas in the comments by Mikael de la Salle and myself).


There is a family of matrix norms that can be defined as follows: given $p \geq 1,k \leq n$,
$$
\|A\|\_{p,k} = \left(\sum\_{i=1}^k \sigma\_i(A)^p\right)^{1/p},
$$
where we denote by $\sigma\_i(A)$ the $i$th [singular value](https://en.wikipedia.org/wiki/Singular_value) of $A$ (taken in nonincreasing order, $\sigma\_1(A)\geq \sigma\_2(A)\geq \dots$). The most famous examples are the [Schatten norms](https://en.wikipedia.org/wiki/Schatten_norm) ($k=n$) and the [Ky Fan norms](https://en.wikipedia.org/wiki/Singular_value_decomposition#Ky_Fan_norms) ($p=1$), but they are matrix norms for each $k$ and $p$ (see e.g. [this blog post](http://www.njohnston.ca/2009/08/ky-fan-norms-schatten-norms-and-everything-in-between/) by Nathaniel Johnston and the references included in it).


A $n\times n$ matrix $A$ is singular iff $\sigma\_n(A)=0$, and this is equivalent to $\|A\_{p,n}\|=\|A\_{p,n-1}\|$ for every $p$. So any pair of norms with $k=n$ and $k=n-1$ works.


