Finding matrices whose multiplication does not change the trace

Question

Let $C$ be an $ntimes n$ matrix. Then find matrices $A$ of size $n times n$ such that $trace(AC)=trace(C)$. One obvious choice for $A$ is the Identity matrix $I$, but are there other choices?
A similar question is, if $A$ and $C$ are $mtimes m$ matrices and $B$ and $D$ are $ntimes n$ matrices such that $trace(C)=trace(D)$ and $trace(AC)=trace(BD)$, then what is the explicit relationship between $A$ and $B$? Here also we can choose $A$ and $B$ to be Identity matrices, but are there other choices of matrices?

Chrystomath · Accepted Answer

The expression $mathrm{trace}(X^*C)$ can be thought of as an inner product between matrices.
So the equation $mathrm{trace}(AC)=mathrm{trace}(C)$ is equivalent to $langle A^*-I,Crangle=0$. So $A^*=I+D$ where $D$ is any matrix 'perpendicular' to $C$. One can create a basis for this space by starting with the elementary matrices $E_1,ldots,E_{n^2}$ and modifying them: $$D_i=E_i-frac{langle E_i,Crangle}{langle C,Crangle} C=E_i-frac{mathrm{trace}(E_i^*C)}{mathrm{trace}(C^*C)}C$$
Hence the answer is $A=I+sum_{i=1}^{n^2-1}alpha_ileft(E_i^*-frac{mathrm{trace}(E_i^*C)}{mathrm{trace}(C^*C)}C^*right)$ for any $alpha_iinmathbb{C}$.
[For $C=0$, then any matrix $A$ is valid.]

Math_Buddy · Answer

The answer to this question is "find it with this possible hint" Choose $A-I=B$ a non invertible matrix such that $BC=0$ then $det(A-I)=0$.
This is answer to the first question.
For 2nd question try this, might work.
$$
A=C=left[begin{array}{ll}
I_{n} & O\
O & O \
end{array}right]
$$
and $B=D=I_{n}$

Finding matrices whose multiplication does not change the trace

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