Triangle inequalities for the operator symmetric modulus

Abstract: We study the operator symmetric modulus (|Z|+|Z^*|)/2 for matrices Z. Several triangle type inequalities are given.

• The paper proves a universal √2 triangle inequality for the operator symmetric modulus over all finite matrix sums and unitarily invariant norms, while noting that optimality remains open.
• For polar Hermitian sums—including Hermitian matrices and involutions—the authors recover constant-1 eigenvalue and norm inequalities, imposing lower bounds on many summed-modulus singular values.
• The paper establishes that the 1/4 correction term in a related inequality for normal matrices is sharp for dimensions n ≥ 3, while the two-dimensional case remains unresolved.

Overview

This paper by Bourin and Lee studies triangle-type inequalities for the symmetric modulus of a matrix, defined for $Z \in M_n$ as

$|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$

where $|Z| = (Z^*Z)^{1/2}$ is the usual (right) modulus. The symmetric modulus is self-adjointly homogeneous and coincides with $|Z|$ when $Z$ is normal, but it lacks the subadditivity properties of the ordinary modulus. The starting point is a recent observation of T. Zhang: the Thompson-type triangle inequality that Zhang proved for the quadratic symmetric modulus $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$ fails for $|\cdot|_{\mathrm{sym}}$; specifically, there exist $A,B \in M_2$ with $$\||A+B|_{\mathrm{sym}}\|_\infty > \||A|_{\mathrm{sym}}\|_\infty + \||B|_{\mathrm{sym}}\|_\infty$$ for the operator norm. The paper's contribution is to supply quantitative substitutes: a universal factor-$\sqrt{2}$ bound valid for all unitarily invariant norms, exact triangle inequalities under structural hypotheses on the summands' sum, and a sharpness result for a related inequality for normal matrices.

A universal $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$0 triangle inequality

The main technical device is a parametric operator inequality obtained from block-matrix positivity arguments. For any matrices $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$1, there exists a unitary $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$2 — namely the unitary factor in the polar decomposition of $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$3 — such that, for every $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$4,

$|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$5

The proof is elementary but effective: the four positive block matrices built from the polar decompositions of the summands are added pairwise, conjugated by the polar unitary of the sum, and then contracted along a scalar parameter to produce the desired Loewner bound. Since $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$6 and likewise for $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$7, this yields a purely symmetric-modulus version, and optimizing over $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$8 (the convex function $|Z|_{\mathrm{sym}} := \frac{|Z| + |Z^*|}{2},$9 is minimized at $|Z| = (Z^*Z)^{1/2}$0) gives the headline results:

• Eigenvalue estimate: for every integer $|Z| = (Z^*Z)^{1/2}$1,

$|Z| = (Z^*Z)^{1/2}$2

via Weyl's inequalities applied to the three-term decomposition of the right-hand side.

• Norm estimate: for every unitarily invariant norm,

$|Z| = (Z^*Z)^{1/2}$3

The authors explicitly leave open whether $|Z| = (Z^*Z)^{1/2}$4 is optimal. Note also that taking traces recovers the classical trace triangle inequality $|Z| = (Z^*Z)^{1/2}$5, so the loss of a constant factor is purely an operator-norm phenomenon.

Polar Hermitian sums: recovering the constant 1

The $|Z| = (Z^*Z)^{1/2}$6 factor can be removed entirely when the sum $|Z| = (Z^*Z)^{1/2}$7 belongs to the class of polar Hermitian matrices, introduced here: matrices whose polar decomposition has a unitary part that is a scalar multiple of a Hermitian unitary. This class contains both Hermitian matrices and involutions ($|Z| = (Z^*Z)^{1/2}$8); for involutions, the fact that the polar unitary is Hermitian follows from uniqueness of the polar decomposition of invertible matrices.

For such sums, the main inequality specializes to one involving only symmetric moduli and a single Hermitian unitary conjugation:

$|Z| = (Z^*Z)^{1/2}$9

with $|Z|$0 Hermitian unitary. Choosing $|Z|$1 and applying Weyl's inequalities gives the exact eigenvalue relation

$|Z|$2

and hence, for every symmetric norm, a genuine triangle inequality with constant 1. A notable corollary: if $|Z|$3 is a Hermitian unitary, then $|Z|$4 for all $|Z|$5. In other words, any decomposition of a symmetry forces at least half of the singular values of the summed symmetric moduli to be at least 1 — a nontrivial constraint on how a unitary involution can be split into summands.

Sharpness for normal matrices

The paper also records a companion inequality for pairs of normal matrices $|Z|$6: there exists a unitary $|Z|$7 such that

$|Z|$8

This follows from the authors' earlier theorem on positive linear maps applied to normal matrices (taking $|Z|$9 and the corner-sum map), and it is the additive analogue of their known Schur-product version, where the constant $Z$0 was already shown to be sharp even in $Z$1. The new observation here concerns sharpness of the additive form: using a previously established example of two Hermitian contractions $Z$2 satisfying $Z$3 with irreducible constant $Z$4, together with the general bound $Z$5 for contractions, they conclude that $Z$6 cannot be improved when $Z$7. Whether $Z$8 is sharp for $Z$9 remains open.

Relation to Thompson's inequality and subadditivity

Section 5 gives an alternative proof of Zhang's quadratic-modulus triangle inequality, $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$0 for some unitaries $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$1, via Thompson's classical inequality $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$2. The key reduction is the equivalence between Thompson's unitary formulation and its contraction formulation, which allows embedding two stacked blocks into a larger space. Applying Thompson's inequality to block matrices of the form $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$3 yields $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$4, and choosing $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$5, $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$6 produces Zhang's result after division by $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$7. The same mechanism underlies the known subadditivity inequality for concave functions $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$8: $|Z|_{qsym} = \big((|Z|^2+|Z^*|^2)/2\big)^{1/2}$9 for positive $|\cdot|_{\mathrm{sym}}$0. Combining these tools, the authors derive a curious exponential variant: for some unitaries $|\cdot|_{\mathrm{sym}}$1,

$|\cdot|_{\mathrm{sym}}$2

obtained by applying the concave function $|\cdot|_{\mathrm{sym}}$3 to Zhang's inequality.

Refinements via the geometric mean

For polar Hermitian sums, the authors also state (without full development) a geometric-mean refinement: there exists a Hermitian unitary $|\cdot|_{\mathrm{sym}}$4 such that

$|\cdot|_{\mathrm{sym}}$5

where $|\cdot|_{\mathrm{sym}}$6 denotes the matrix geometric mean. By the extremal characterization of $|\cdot|_{\mathrm{sym}}$7, this strengthens the constant-1 norm inequality to a weak log-majorisation:

$|\cdot|_{\mathrm{sym}}$8

placing the result in the same hierarchy as the classical log-majorisation $|\cdot|_{\mathrm{sym}}$9 for normal $A,B \in M_2$0.

Limitations and open questions

Several points are left unresolved or rest on unverified assumptions. The optimality of the constant $A,B \in M_2$1 in the general symmetric-modulus triangle inequality is posed as a question rather than proved. For the normal-matrix inequality, sharpness of $A,B \in M_2$2 is established only for $A,B \in M_2$3, and the best constant in dimension two is unknown. The geometric-mean refinements are stated but not treated in depth, as the authors prioritize sharp-constant triangle inequalities. Finally, the paper poses a broader question about expansive matrices ($A,B \in M_2$4): given a decomposition $A,B \in M_2$5, what lower bounds hold for the middle singular values $A,B \in M_2$6 and $A,B \in M_2$7? The motivating example in $A,B \in M_2$8 shows that decomposing a Hermitian unitary imposes $A,B \in M_2$9 on the summed moduli, suggesting analogous constraints should exist in general, but no theorem is provided.

Conclusion

The paper establishes that although the symmetric modulus fails the operator-norm triangle inequality outright, a uniform $$\||A+B|_{\mathrm{sym}}\|_\infty > \||A|_{\mathrm{sym}}\|_\infty + \||B|_{\mathrm{sym}}\|_\infty$$0 substitute holds for all unitarily invariant norms and arbitrary finite families of matrices, and the sharp constant 1 is recovered whenever the sum is polar Hermitian — in particular for involutions and Hermitian sums. Alongside a sharpness proof for the $$\||A+B|_{\mathrm{sym}}\|_\infty > \||A|_{\mathrm{sym}}\|_\infty + \||B|_{\mathrm{sym}}\|_\infty$$1-constant inequality for normal sums in dimensions $$\||A+B|_{\mathrm{sym}}\|_\infty > \||A|_{\mathrm{sym}}\|_\infty + \||B|_{\mathrm{sym}}\|_\infty$$2, and a streamlined derivation of Zhang's quadratic-modulus theorem from Thompson's inequality, the work maps out precisely where the symmetric modulus behaves like a genuine modulus and quantifies the deviation where it does not.