Minimum of $|\det(X+iC)|$

Question

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of:

$$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$

When $n=1$ it's clear that the minimum value equal $|C|$, however it seems like with $n\ge2$ the minimum value is exactly zero, but I don't know how to prove it.

I have found out that this is relating to the minimum of $\det(I+Q^2)$ with $I$ is identity and $Q$ is arbitrary real matrix. If you can evaluate either:

• $|\det(X+iC)|$

• $\det(I+Q^2)$

it would be appreciated!

Answer 1

For any $C$, one can always find an $X$ such that $\det(X+iC)=0$ : if $C$ is not invertible, simply take $X=0$. If $C$ is invertible, take $X=CA$ where $A$ is any real matrix having a two-dimensional invariant subspace ${\sf span}(v_1,v_2)$ with $Av_1=v_2$, $Av_2=-v_1$ (so that $\pm i$ are eigenvalues of $A$). Then

$$ \det(X+iC)=\det(C(A+iI_n))=\det(C)\times 0=0. $$

Update I’ve just realized that my second case construction works in all cases, there’s no need to assume that $C$ is invertible.

Answer 2

If we assume true that the minimum is zero it means that we can minimize separately the real part and the imaginary part. Let n=2, knowing that $\det(X+iC)=\det(X)+i(\det(x_1,c_2)+\det(c_1,x_2))-\det(C)$ then it means that $\det(X)=\det(C)$ and $\det(x_1,c_2)=\det(x_2,c_1)$ where $x_i$ and $c_i$ represents the i-st columns of the matrices $X$ and $C$ respectively. If $\det(C)=0$ we would have that $X=C$ would be a minimizer with value $0$, so we can assume that $\det(C)\neq0$. We end the argument taking $X=(c_2,-c_1)$.

If $n>2$ we can note that taking $X=(c_2,-c_1,c_3,\dots,c_n)$ will give us: $$X+iC=(c_2+ic_1,-c_1+ic_2,\dots)=(c_2+ic_1,i(c_2+ic_1),\dots)$$ Which is clearly singular.

On a same note $\det(I+Q^2)=0$ if on some two dimensional space $Q$ is a rotation of a $\frac{\pi}{2}$ angle. Again if $n=2$ $$Q=\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right)$$ then $Q^2=-I$. If $n>2$ then as written above it is needed that $\exists v_1, v_2\in\mathbb{R}^n$ s.t. $\|v_1\|=\|v_2\|$ and $Qv_1=v_2$ and $Qv_2=-v_1$.