Q:

How to use a non-const member function inside a const member function?

Following are a simple C++ struct and a class for 3D vector and 3x3 matrix. As you can see the subscript operator is overloaded for both of them. I return a reference to the data members from them in order to be able to assign to those members like vec[0] = 5.0;, mat[0] = vec3(2.0); and mat[0][0] = 3.0; etc. Now I declare another operator for mat3, the plus(+) operator which should not modify the matrix itself but should return a new matrix as the sum. How can I make this non-modifying + operator as const?
Vector
struct vec3 {
    typedef double value_type;
    vec3::value_type x, y, z;

    vec3(const vec3::value_type x,
         const vec3::value_type y,
         const vec3::value_type z)
    : x{x}, y{y}, z{z} {}

    vec3(const vec3::value_type w)
    : vec3(w, w, w) {}

    vec3()
    : vec3(0.0) {}

    // this works because [] operator is not used
    vec3 operator+(const vec3& v) const {
        return vec3(this->x + v.x, this->y + v.y, this->z + v.z);
    }

    vec3::value_type& operator[](const std::size_t index) {
        switch(index) {
        case 0:
            return x;
        case 1:
            return y;
        case 2:
            return z;
        default:
            throw std::invalid_argument("vec3 supports upto 3(0-2) elements");
        }
    }
};

Matrix
class mat3 {
    std::array<vec3,3> val;
public:
    using vtype = vec3::value_type;
    mat3(const vtype v00, const vtype v01, const vtype v02,
         const vtype v10, const vtype v11, const vtype v12,
         const vtype v20, const vtype v21, const vtype v22) {
        val[0][0] = v00; val[0][1] = v10; val[0][2] = v20;
        val[1][0] = v01; val[1][1] = v11; val[1][2] = v21;
        val[2][0] = v02; val[2][1] = v21; val[2][2] = v22;
    }

    mat3(const vtype m[3][3]) {
        for(std::size_t i = 0; i < 3; ++i) {
            for(std::size_t j = 0; j < 3; ++j) {
                val[i][j] = m[j][i];
            }
        }
    }

    mat3(const vtype v)
    : mat3(v, v, v,
           v, v, v,
           v, v, v) {}

    mat3()
    : mat3(0.0) {}

    // how to make it `const`
    mat3 operator+(const mat3& m) {
        mat3 t;
        for(std::size_t i = 0; i < 3; ++i) {
            t[i] = val[i] + m[i];
        }
        return std::move(t);
    }

    vec3& operator[](const std::size_t index) {
        switch(index) {
        case 0:
            return val[0];
        case 1:
            return val[1];
        case 2:
            return val[2];
        default:
            throw std::invalid_argument("mat3 supports upto 3(0-2) vec3");
        }
    }
};

A:

You could (and should) add a const version for operator[], then you can make operator+ const, the const version of operator[] will be invoked in it.
vec3& operator[](const std::size_t index) {
    ...
}
const vec3& operator[](const std::size_t index) const {
~~~~~                                           ~~~~~
    ...
}

Note you can overload const and non-const member functions.
See const-, volatile-, and ref-qualified member functions (emphasis mine)

A non-static member function can be declared with a const, volatile,
  or const volatile qualifier (this qualifier appears after the name of
  the function in function declaration). Differently cv-qualified
  functions have different types and so may overload each other.
In the body of a cv-qualified function, the this pointer is
  cv-qualified, e.g. in a const member function, only other const member
  functions may be called normally. (A non-const member function may
  still be called if const_cast is applied or through an access path
  that does not involve this.)

