TY - JOUR
T1 - On the block structure and frobenius normal form of powers of matrices
AU - Albaidani, Mashael M.
AU - McDonald, Judi J.
N1 - Publisher Copyright:
© 2019, International Linear Algebra Society. All rights reserved.
PY - 2019
Y1 - 2019
N2 - The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this article, conditions on a matrix A and the power q are provided so that for any invertible matrix S, if S−1Aq S is block upper triangular, then so is S−1AS when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. The article concludes by applying the block upper triangular powers result to the cone Frobenius normal form of powers of a eventually cone nonnegative matrix.
AB - The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this article, conditions on a matrix A and the power q are provided so that for any invertible matrix S, if S−1Aq S is block upper triangular, then so is S−1AS when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. The article concludes by applying the block upper triangular powers result to the cone Frobenius normal form of powers of a eventually cone nonnegative matrix.
KW - Block upper triangular matrices
KW - Cones
KW - Eventually nonnegative matrices
KW - Fields
KW - Frobenius normal form
UR - http://www.scopus.com/inward/record.url?scp=85073273421&partnerID=8YFLogxK
U2 - 10.13001/1081-3810.3955
DO - 10.13001/1081-3810.3955
M3 - Article
AN - SCOPUS:85073273421
SN - 1537-9582
VL - 35
SP - 297
EP - 306
JO - Electronic Journal of Linear Algebra
JF - Electronic Journal of Linear Algebra
IS - 1
M1 - 18
ER -