Let D be a diagonal matrix and A a Hermitian one. Is there a nontrivial way to calculate the determinant of A from the determinant of A+D and the entries of D?
It can be assumed that the diagonal entries of A are all zeros.
Thankyou very much.
Let D be a diagonal matrix and A a Hermitian one. Is there a nontrivial way to calculate the determinant of A from the determinant of A+D and the entries of D?
It can be assumed that the diagonal entries of A are all zeros.
Thankyou very much.
Take $A+D=\begin{pmatrix} 1& a& b\\ a& 1& c\\ b& c& 1 \end{pmatrix}$ with $a$, $b$, $c$ real.
Now, the entries of $D$ are 1 and the determinant of $A+D$ is $1+2abc-a^2-b^2-c^2$.
You want to recover $\det A=2abc$.
If you regard $a=b=1$ and $c=0$, then $\det A+D=-1$ and $\det A=0$.
But if you regard $a=b=1$ and $c=2$, then you also get $\det A+D=-1$, but $\det A=4$.
This example shows that your information is not sufficient to distinguish the two cases. So, in general it is impossible to recover the determinant of the original matrix $A$.