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Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\mathcal{N}}$ be the nuclear norm in $\mathcal{M}_n(\mathbb{C})$. How to compute the distance from the given matrix $\varphi$ to the subspace $Z$ with respect to the nuclear norm, i.e. the quantity

$d_{\mathcal{N}}(\varphi,Z):=\inf\{\|\varphi-zI\|_{\mathcal{N}}\colon\,z\in\mathbb{C}\}$.

Or at least how to find a complex number $z_0\in\mathbb{C}$ that realizes the above infimum?

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    $\begingroup$ In other words the trace-class norm of $\varphi$, i.e. the sum of the square roots of the eigenvalues of the matrix $\varphi^*\varphi$. $\endgroup$
    – Krzysztof
    Commented Sep 19 at 11:55
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    $\begingroup$ Just to give you a term to Google, you are looking for matrices that are "Birkhoff-James orthogonal" to $I$ with respect to the trace norm. As Mark L. Stone said in an answer, you can solve this numerically via semidefinite programming, but I wouldn't be surprised if there's an explicit answer known at least if $\phi$ is, e.g., Hermitian. $\endgroup$
    – Nathaniel Johnston
    Commented Sep 19 at 14:06
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    $\begingroup$ Further to @NathanielJohnston's final comment: in the Hermitian case $\varphi$ is unitarily equivalent to a diagonal matrix with real entries $\lambda_1,\dots,\lambda_n$ and then you are seeking $z\in {\mathbb R}$ minimizing $\sum_{j=1}^n |\lambda_j- z|$. $\endgroup$
    – Yemon Choi
    Commented Sep 19 at 16:12

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Edit: As discussed in comments by @Yemon Choi and @Nathaniel Johnston , there is a simpler solution if $\psi$ is Hermitian. So the remainder of my post addresses the non-Hermitian case, although it is valid as well in the Hermitian case.

The "Sum of singular values" subsection of section 6.2.4 "Singular value optimization" of the Mosek Modeling Cookbook shows two Linear SDP formulations (which are duals of each other) of this problem. These formulations allow the problem to be solved numerically as a convex optimization problem with an SDP solver.

Alternatively, CVX can be used, which has a norm_nuc function which does the SDP formulation under the hood, and calls an SDP solver, such as Mosek, SeDuMi, or SDPT3, to solve it.

cvx_begin
variable z complex
minimize(norm_nuc(psi - z*eye(n))
cvx_end
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