Does $Ax+Bx=\lambda x$ imply $\langle Ax ,x \rangle + \langle Bx, x \rangle=\langle\lambda x, x \rangle$?

Question

Let $A$ and $B$ be real matrices. Let $\lambda$ be an eigenvalue of $A+B$. Let $x$ be a corresponding eigenvector

Then $Ax+ Bx= \lambda x$.

My question is:

Is it valid to take the inner product on both side of the equation here?

Like $ \langle Ax ,x \rangle + \langle Bx, x \rangle = \langle \lambda x, x \rangle $. Is this valid?

In general, can we always take the inner product on both sides of an equation?

Thanks!

Answer 1

Yes. As inner product is additive in each component. But you have to be careful while dealing with homogeneity as inner product is conjugate linear. But in real inner product it's linear in both components i.e a real inner product is a bilinear map.

Here we only need additivity, so it's fine for any inner product space.

$\langle \lambda x,x\rangle= \langle Ax+ Bx,x \rangle=\langle Ax, x\rangle+\langle Bx,x \rangle$