symbolic derivation - toy example: OLS regression

I want to apply matlabs symbolic toolbox to find solutions to some matrix calculus problems.

As a toy example I would like to derive the optimal weights in an ordinary least squares regression problem. Linear Regression as well as it's analytic solution are widely known and thus hopefully provide a relatable problem case.

Derivation by Hand

Starting from the Least Squares Error as minimization objective it follows:

We derivate with respect to the weights:

Set to zero and solve for w:

Derivation using the Symbolic Toolbox

Now how can this be done using the symbolic toolbox?

We can specify y, w and X as

X = sym('X', [3 5], 'real')
y = sym('y', [5 1], 'real')
w = sym('w', [3 1], 'real')

It would be favourable to work in arbitrary dimensions p and q. Is this possible? (No documentation found)

State the Least Squares Error:

E1(w) = norm(y - X'*w)
E2(w) = (y - X'*w)' * (y - X'*w)
E3(w) = y'*y- 2 * w'*X*y + w'*X*X'*w

However if I subtract one from another there are terms remaining, but we know they should cancle out. Where am I going wrong here?

Continuing with the gradient with respect to the vector w gives:

Edw = gradient(E3,w)

Setting it to zero and solve for w:

solve(Edw == 0, w)

This returns a struct with 3 fields, each of shape 1x1 with no values.

But I know the solution should be .

How would you approach this problem?

More generally is it possible to work on a matrix representation level rather than on it's elementwise formulation? In the derivation I did by hand I did not have go into the details of expanding out every matrix multiplication either. It is much cleaner that way.

Thank you in advance