# Polynomially Isometric Matrices in Low Dimensions

@article{Brooks2021PolynomiallyIM, title={Polynomially Isometric Matrices in Low Dimensions}, author={Cara D. Brooks and Alberto A. Condori and Nicholas Seguin}, journal={The American Mathematical Monthly}, year={2021}, volume={128}, pages={513 - 524} }

Abstract Given two d × d matrices, say A and B, when do p(A) and p(B) have the same “size” for every polynomial p? In this article, we provide definitive results in the cases d = 2 and d = 3 when the notion of size used is the spectral norm.

## References

SHOWING 1-10 OF 21 REFERENCES

### Pseudospectra do not determine norm behavior, even for matrices with only simple eigenvalues

- Mathematics
- 2011

### A Resolvent Criterion for Normality

- MathematicsAm. Math. Mon.
- 2018

It is proved that a certain distance formula is in fact sufficient for normality and it is demonstrated that the spectrum of a matrix can be used to recover the spectral norm of its resolvent precisely when the matrix is normal.

### A complete set of unitary invariants for 3×3 complex matrices

- Mathematics
- 1962

is a complete set of unitary invariants for any nXn complex matrix A. The author was able to improve this result by demonstrating in [3] that for n fixed but arbitrary, there is always a subset of…

### A minimal polynomial basis of unitary invariants of a square matrix of the third order

- Mathematics
- 1968

A simple method of constructing a minimal polynomial basis and a minimal complete system of unitary invariants of a square matrix of the third order is presented.

### Do the Pseudospectra of a Matrix Determine its Behavior

- Mathematics
- 1993

It is shown that matrix behavior cannot be inferred from the norm of the resolvent (Theorem 2), and whether the gap between these two kinds of information may be quantiiable.

### Super‐identical pseudospectra

- Mathematics
- 2009

The complex N × N matrices A and B are said to have super‐identical pseudospectra if, for each z ∈ ℂ, the singular values of A − zI are the same as those of B − zI. We explore this condition and its…

### On the closure of the numerical range of an operator

- Mathematics
- 1967

If T is a bounded linear mapping (briefly, operator) in a Hilbert space SC, the numerical range of T is the set W(T) = { (Tx, x): x|| ==1}; thus W(T) is convex [8, p. 131], and its closure cl[W(T)]…

### Linear Algebra Done Right

- Mathematics
- 1995

-Preface for the Instructor-Preface for the Student-Acknowledgments-1. Vector Spaces- 2. Finite-Dimensional Vector Spaces- 3. Linear Maps- 4. Polynomials- 5. Eigenvalues, Eigenvectors, and Invariant…