I have a question related with singular values of matrix sums.

Let's assume I have matrices $A$, $B$, and $D$ (positive, semi-definite) where $D = A + B$. For singular values of $D$, I know that

$$ λ(D) \le ∑λ(A)+∑λ(B). $$

However, if I want to calculate the singular values of $D = A + cB$, where $c$ is a constant (real number) other than just $1$, does the assumption above still hold?

Thus, can I assume that for the singular values of $D = A + cB$ the following holds

$$ ∑λ(D) \le ∑λ(A) + c∑λ(B). $$

and$A + c B$ are all positive semidefinite, the singular values are just the eigenvalues and their sum is the trace. Then of course $\sum \lambda(A+cB) = \sum \lambda(A) + c \sum \lambda(B)$. You only need an inequality when $A+cB$ might not be positive semidefinite, and that inequality will require $c \ge 0$. $\endgroup$