Consider two real-valued positive semi-definite matrices $A$ and $B$, and suppose $C = A + B$. I am interested in proving that $\det(C) \geq \det(A)$.
I had heard through a colleague that the eigenvalues of $C$ are each bigger than either the eigenvalues of $A$ or the eigenvalues of $B$. Is there a name for this result?
If I could get a name for this result, I could argue that $\det(C) = \prod_{i=1}^n \lambda_i \geq \prod_{i=1}^n \gamma_i = \det(A)$, where $\gamma_i$ is the $i$'th eigenvalue of $A$ and $\lambda_i$ is the $i$'th eigenvalue of $C$.
Is there a name for this result, or a paper or book I could cite?
Best Answer
We use the result given in this question: if $A$ and $B$ are two symetric positive definite matrices then $\det(A+B)>\det(A)$.
In your case, $A$ and $B$ are not necessary positive definite, but $A+B+2k^{-1}Id$ is so $\det(A+B+2k^{—1})> \det(A+k^{-1})$ and taking the limit $k\to +\infty$, $\det(A+B)\geq\det(A)$.