Eigenvalues of the addition of a symmetric matrix and the identity matrix

Question

Let $X \in \mathbb{R}^{m \times m}$ be a symmetric matrix. What are the eigenvalues of $X+I$, where $I$ is the identity matrix?

I need to show that the eigenvalues are now all $+1$. The explanation decomposes $X$ and $I$. $$X = U\Sigma U^{T}$$ $$I = UIU^{T}$$ $$X + I = U(\Sigma+I)U^{T}$$

Why is the the third step true?

Answer 1

$$X = U\Sigma U^{T}$$ $$I = UIU^{T}$$

Adding the two equations. $$X + I = U \Sigma U^T+UIU^T=U(\Sigma U^T+IU^T)=U(\Sigma+I)U^T$$

where in the second equation, I factorize $U$ out, and in the last equation, I factorize $U^T$ out.