Let $\mathbb{N}$ be the set of non-negative integers.  Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(\mathbb{N})$ and $\Vert A \Vert$ the [operator norm][1]. Recall that $\Vert A \Vert^2$ is the largest eigenvalue of $A^*A$. 

If $A = uv^*$ with $u,v \in \mathbb{N}^n$ then $A^*A = v u^*uv^*$, so that  $\Vert A \Vert^2 = \Vert u \Vert^2\Vert v \Vert^2$ and $E_nE_n \subseteq F_n$.   

[This answer][2] and [Lagrange's four square theorem][3] imply that $E_n = F_n$ $\forall n \neq 3$, whereas $E_3 \subsetneq F_3$ because, by [Legendre's three-square theorem][4], $E_3 = \mathbb{N} \setminus \{ 4^a(8b+7) \ | \ a,b \in \mathbb{N} \} \subsetneq E_3E_3$.

The investigation below shows that $F_3$ contains every non-negative integer less than $2 \cdot 10^6$.

>**Question**: Does the form $\Vert A \Vert^2$ cover every non-negative integer for $A \in M_3(\mathbb{N})$?

*Remark:* It is proved [here][5] for $A \in M_3(\mathbb{Z})$.

__________
**Investigation**  

Observe that $L:= \mathbb{N} \setminus E_3E_3$ is the set of positive integers $8a-1$ whose prime factors are of the form $8b \pm 1$. The first elements of $L$ are $7, 23, 31, 47, 71, 79, 103, 119, 127, 151, 167, \dots$  

Take $A=(u_1,u_2,u_3)$ with $u_i \in \mathbb{N}^3$. Then the characteristic polynomial of $A^*A$ is
$$P(x) = x^3-\left(\sum_{i=1}^3 \Vert u_i \Vert^2 \right)x^2 + \left( \sum_{i<j} \Vert u_i \times u_j \Vert^2 \right)x - \left( u_1 \cdot (u_2 \times u_3) \right)^2 $$ with $u \times v$ the [cross product][6] and $u \cdot v$ the [dot product][7]. 

Assume that the vectors $u_1, u_2, u_3$ are linearly dependent. Then

$$\Vert A \Vert^2 = \frac{1}{2} \left(  \sum_{i=1}^3 \Vert u_i \Vert^2
 + \sqrt{\left( \sum_{i=1}^3 \Vert u_i \Vert^2 \right)^2 -4 \sum_{i<j} \Vert u_i  \times u_j \Vert^2} \right)$$

Assume moreover that $u_3=0$, and then observe that
$$\Vert A \Vert^2 = \frac{1}{2} \left(  \Vert u_1 \Vert^2 + \Vert u_2 \Vert^2
 + \sqrt{\left( \Vert u_1 \Vert^2 - \Vert u_2 \Vert^2 \right)^2 + 4 (u_1  \cdot u_2)^2} \right).$$
If moreover $\Vert u_1 \Vert = \Vert u_2 \Vert$ then $\Vert A \Vert^2 = \Vert u_1 \Vert^2 + (u_1  \cdot u_2) = \frac{1}{2} \Vert u_1+u_2 \Vert^2 $, and we know by [this post][8] that this form covers every odd less than $90000$, except those in $\{ 5, 23, 29, 65, 167 \}$, but its intersection with $L$ is just $\{23,167\}$ whereas   
 $$ 23 = \left\| \pmatrix{0&2&0\\ 1&4&0 \\ 2&1&0} \right\|^2 \  \text{ and } \ 167 = \left\| \pmatrix{0&7&0\\ 2&5&0 \\ 8&7&0} \right\|^2.$$
It follows that every non-negative integer less than $90000$ is covered.  

If $u_1=(a,b,c)$, $u_2 = (b,c,a)$ and $u_3=0$, then $2\Vert A \Vert^2 = (a+b)^2+(b+c)^2+(c+a)^2$.  
Observe (after [this comment][9]) that we are reduced to show that $\forall n > 90000$, if $n \in L$ then $2n$ is of the form $x^2+y^2+z^2$ (which is true by Legendre's three-square theorem) with the *stronger* assumption that $x,y,z$ are the integer sides of a triangle (i.e. $x \le y \le z \le x+y $). It is checked below for $2543<n<2 \cdot 10^6$. Then every non-negative integer less than $2 \cdot 10^6$ is covered.  

*Remark*: This *stronger* version of Legendre's three-square theorem is suspected to be true for every $2n$ with $n$ odd greater than $5969$ (see [this post][11]), and in general every *large* element of $E_3$.
_____
*Computation*

    sage: ModelOutEE(1,2000000)
    [23, 167, 239, 479, 623, 1031, 1439, 1751, 2543]

_______
**Code**

    cpdef StrongLegendre(int i):
    	cdef int n,a,b,c,j
    	n=isqrt(i)
    	for a in range(n+1):
    		for b in range(a+1):
    			j=i-a**2-b**2
    			if j>=0:
    				c=isqrt(j)
    				if c**2==j:
    					if c<=a and a<=b+c:
    						return True
    					if c>a and c<=a+b:
    						return True
    	return False
    
    cpdef is_EE(int i):
    	cdef int a,l
    	cdef list f
    	cdef tuple j
    	if not Integer(i).mod(8)==7:
    		return True
    	b=0
    	f=list(factor(i))
    	for j in f:  
    		a=j[0]
    		if not Integer(a).mod(8) in [1,7]:
    			return True
    	return False
    
    cpdef ModelOutEE(int r1, int r2):
    	cdef int i
    	cdef list L
    	L=[]
    	for i in range(r1,r2):
    		if not is_EE(i):
    			if not StrongLegendre(2*i):
    				L.append(i)
    	return L


  [1]: https://en.wikipedia.org/wiki/Matrix_norm#Matrix_norms_induced_by_vector_norms
  [2]: https://math.stackexchange.com/a/2963285/84284
  [3]: https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem
  [4]: https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem
  [5]: https://math.stackexchange.com/a/2967989/84284
  [6]: https://en.wikipedia.org/wiki/Cross_product
  [7]: https://en.wikipedia.org/wiki/Dot_product
  [8]: https://mathoverflow.net/q/313675/34538
  [9]: https://math.stackexchange.com/questions/2975032/odd-of-the-form-a2b2c2abacbc#comment6140448_2975032
  [11]: https://math.stackexchange.com/q/2975032/84284