MATLAB: Fix inv warning in matlab

Question

I use this code:

b=inv(A'*A)*A'*y; 

Matlab gives warning. never use inv to solve linear system

How can I fix it?

Answer 1

b=inv(A'*A)*A'*y; 

Multiply both sides on the left by A'*A :

(A'*A) * b = (A'*A) * inv(A'*A) * A' * y

and for any invertable matrix, X * inv(X) is the identity matrix so (A'*A) * inv(A'*A) cancels out to identity, so

(A'*A) * b = A' * y

Multiply both sides on the left by inv(A'):

inv(A') * A' * A * b = inv(A') * A' * y

and inv(A') * A' cancels to the identity on both sides, so

A * b = y

Multiply by inv(A) on the left on both sides:

inv(A) * A * b = inv(A) * y

inverse cancels to identity, so

b = inv(A) * y

Now use the \ operator:

b = A \ y;

The above mathematics might not strictly apply if A is not a square matrix.