Consequences of Matrices Sharing Eigenvalues and Eigenvectors
Abstract
While studying for a linear algebra final, the first named author prepared some test questions for herself to see how well she understood the material, and asked the second named author: \emph{If A and AT have the same eigenvalues and eigenvectors, is A a symmetric matrix?} We show how this excellent question is a great springboard to related questions, in particular when do equal eigenvalues and eigenvectors imply the matrices are, if not equal, at least closely related (such as similar or the transpose/complex conjugate transpose of each other)? The answer depends on how we interpret the question, and provides a great opportunity to talk about creating good questions. In particular, we characterize matrices A for which the transpose or conjugate transpose shares the same eigenvectors (regardless of eigenvalues) and, for each eigenvalue, the same eigenpair (equivalently, the same eigenspace). Thus, a square matrix A is Hermitian if and only if A* has the same eigenpairs as A; moreover, if A is a real matrix with real eigenvalues and AT has the same eigenvectors as A, then A is symmetric.
Related
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BibTeX
@misc{miller2026consequences,
title = {{Consequences of Matrices Sharing Eigenvalues and Eigenvectors}},
author = {Kayla Miller and Steven J. Miller and Fuzhen Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10578},
eprint = {2609.10578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10578},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Consequences of Matrices Sharing Eigenvalues and Eigenvectors
AU - Kayla Miller
AU - Steven J. Miller
AU - Fuzhen Zhang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.10578
UR - https://arxiv.org/abs/2609.10578
AB - While studying for a linear algebra final, the first named author prepared some test questions for herself to see how well she understood the material, and asked the second named author: \emph{If A and A^T have the same eigenvalues and eigenvectors, is A a symmetric matrix?} We show how this excellent question is a great springboard to related questions, in particular when do equal eigenvalues and eigenvectors imply the matrices are, if not equal, at least closely related (such as similar or the transpose/complex conjugate transpose of each other)? The answer depends on how we interpret the question, and provides a great opportunity to talk about creating good questions. In particular, we characterize matrices A for which the transpose or conjugate transpose shares the same eigenvectors (regardless of eigenvalues) and, for each eigenvalue, the same eigenpair (equivalently, the same eigenspace). Thus, a square matrix A is Hermitian if and only if A^* has the same eigenpairs as A; moreover, if A is a real matrix with real eigenvalues and A^T has the same eigenvectors as A, then A is symmetric.
ER -