Properties of Matrix Norms

• theorem: scalar-valued vector function rotationally invariant iff function of Euclidean norm
• corollary: norm rotationally invariant iff scalar multiple of Euclidean norm
• corollary: for matrix norms induced by vector norms
  • left rotationally invariant iff codomain uses Euclidean norm
  • right rotationally invariant iff domain uses Euclidean norm
• proposition: spectral norm is the only two-sided rotationally invariant norm
  • and spectral norm is two-sided unitarily invariant
• proposition: for matrix norms induced by inner product
  • one-sided rotational invariance (any) implies scalar multiple of Frobenius norm implies unitarily invariant
• theorem:
  • left unitarily invariant matrix norm iff depends only on row Gram matrix
  • right unitarily invariant matrix norm iff depends only on (column) Gram matrix
• theorem (Von Neumann): matrix norm unitarily invariant on both sides iff norm equals some symmetric gauge function of singular values
  • symmetric gauge function: norm axioms + invariant under elementwise absolute value and permutation
  • special case: all Schatten p-norms

Uniqueness of the Spectral Norm

From: (user1551, 2019)

The spectral norm (Schatten \(\infty\)-norm, or operator norm \(\Vert \cdot \Vert_{\ell_2 \to \ell_2}\)) possesses a remarkable uniqueness property. It is the only matrix norm on \(\mathbb{R}^{n \times n}\) that satisfies a specific set of conditions related to orthogonal transformations and submultiplicativity.

Theorem. Uniqueness of the Spectral Norm

Let \(\Vert \cdot \Vert\) be a matrix norm on \(\mathbb{R}^{n \times n}\). If this norm satisfies the following three conditions:

1. Submultiplicativity: \(\Vert AB \Vert \le \Vert A \Vert \Vert B \Vert\) for all \(A, B \in \mathbb{R}^{n \times n}\).
2. Normalization for Orthogonal Matrices: \(\Vert H \Vert = 1\) for any orthogonal matrix \(H \in O(n)\).
3. Left-Orthogonal Invariance: \(\Vert HA \Vert = \Vert A \Vert\) for any orthogonal matrix \(H \in O(n)\) and any matrix \(A \in \mathbb{R}^{n \times n}\).

Then, \(\Vert A \Vert = \sigma_{\max}(A) = \Vert A \Vert_2\) for all \(A \in \mathbb{R}^{n \times n}\).

References

1. 06 02 Matrix Norms and Unitary Matrices. Retrieved June 6, 2025, from https://nla.skoltech.ru/archive/2019/lectures/html/06%2002%20Matrix%20norms%20and%20unitary%20matrices.html
2. Dokmanić, I., & Gribonval, R. (2017). Beyond Moore-Penrose Part I: Generalized Inverses That Minimize Matrix Norms (Number arXiv:1706.08349). arXiv. https://doi.org/10.48550/arXiv.1706.08349
3. Domon, M., Sano, T., & Toba, T. Left Unitarily Invariant Norms on Matrices.
4. Higham, N. (2020). What Is the Polar Decomposition? In Nick Higham. https://nhigham.com/2020/07/28/what-is-the-polar-decomposition/
5. Higham, N. (2021). What Is a Unitarily Invariant Norm? In Nick Higham. https://nhigham.com/2021/02/02/what-is-a-unitarily-invariant-norm/
6. Horn, R. A., & Johnson, C. R. (2017). Matrix Analysis (Second edition, corrected reprint). Cambridge University Press.
7. MIRSKY, L. (1960). SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS. The Quarterly Journal of Mathematics, 11(1), 50–59. https://doi.org/10.1093/qmath/11.1.50
8. Saluev, T., & Sitdikov, I. (2015). Generic Properties and a Criterion of an Operator Norm. Linear Algebra and Its Applications, 485, 1–20. https://doi.org/10.1016/j.laa.2015.07.020
9. user1551. (2019). Answer to "What Are Some Lesser Known Unitary Invariant Norms for Matrices?". In Mathematics Stack Exchange. https://math.stackexchange.com/a/3311122