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

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 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 and $u \cdot v$ the dot product.

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 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) 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), 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