Vectorization not working in Matlab - Matrix dimensions do no agree?

I want to multiply elements of a matrix T against elements of two vectors vec_1 and vec_2, and sum everything up. Using nested for loops, I can do it like this:

T = eye(3);
vec_1 = [4,5,6];
vec_2 = [7,8,9];

tot = 0;
for m=1:3
    for n=1:3
        tot = tot + T(m,n) .* vec_1(m) .* vec_2(n);
    end
end

I wanted to make it faster using vectorization so I tried the following.

T = eye(3);
vec_1 = [4,5,6];
vec_2 = [7,8,9];

f = @(m,n) T(m,n) .* vec_1(m) .* vec_2(n);

[M, N] = meshgrid(1:3,1:3);
tot = sum(f(M,N),'all');

However, this doesn't work and I get the error 'Matrix dimensions must agree.' From debugging it, the problem is due to T being evaluated using M and N. Instead of returning a 3x3 matrix as I expected, T(M,N) returns a 9x9 matrix. How can I fix this code so I can use vectorization instead of nested for loops for this task?

[Stephen23]

As DGM shows, you need to replace T(m,n) with T.

Answer 1

[DGM]

It can be simpler than that.

% original
T = eye(3);
vec_1 = [4,5,6];
vec_2 = [7,8,9];

tot = 0;
for m=1:3
    for n=1:3
        tot = tot + T(m,n) .* vec_1(m) .* vec_2(n);
    end
end
tot

tot = 122

% alternatively
T = eye(3);
vec_1 = [4,5,6];
vec_2 = [7,8,9];

tot = sum(T .* vec_1.' .* vec_2,'all')

tot = 122

If T is always an identity matrix, then it simplifies further

tot = vec_1*vec_2.'

tot = 122