[1] C. Bu and Y. Zhou, Involutory and s+1 potency of linear combinations of a tripotent
matrix and an arbitrary matrix, J. Appl. Math. Inform. 29 (1–2), 485-495, 2011.
[2] X. Liu, J. Benítez and M. Zhang, Involutiveness of linear combinations of a quadratic
or tripotent matrix and an arbitrary matrix, Bull. Iranian Math. Soc. 42 (3), 595-610,
2016.
[3] H. Özdemir and T. Petik, On spectra of some matrices derived from two quadratic
matrices, Bull. Iranian Math. Soc. 39 (2), 225-238, 2013.
[4] H. Özdemir and M. Sarduvan, Notes on linear combinations of two tripotent, idempotent,
and involutive matrices that commute, An. Ştiint. Univ. “Ovidius" Constanta
Ser. Mat. 16, 83-90, 2008.
[5] T. Petik, H. Özdemir and J. Benítez, On the spectra of some combinations of two
generalized quadratic matrices, Appl. Math. Comput. 268, 978-990, 2015.
[6] T. Petik, M. Uç and H. Özdemir, Generalized quadraticity of linear combination of
two generalized quadratic matrices, Linear Multilinear Algebra 63 (12), 2430-2439,
2015.
[7] M. Sarduvan and H. Özdemir, On linear combinations of two tripotent, idempotent,
and involutive matrices, Appl. Math. Comput. 200 (1), 401-406, 2008.
[8] M. Sarduvan and N. Kalaycı, On idempotency of linear combinations of a quadratic
or a cubic matrix and an arbitrary matrix, Filomat 33 (10), 3161-3185, 2019.
[9] M. Tošić, On some linear combinations of commuting involutive and idempotent matrices,
Appl. Math. Comput. 233, 103-108, 2014.
[10] M. Uç, H. Özdemir and A.Y. Özban, On the quadraticity of linear combinations of
quadratic matrices, Linear Multilinear Algebra 63 (6), 1125-1137, 2015.
[11] M. Uç, T. Petik and H. Özdemir, The generalized quadraticity of linear combinations
of two commuting quadratic matrices, Linear Multilinear Algebra 64 (9), 1696-1715,
2016.
[12] Y. Wu, K–potent matrices–construction and applications in digital image encryption,
Recent Advances in Applied Mathematics, AMERICAN-MATH’10 Proceedings of
the 2010 American Conference on Applied Mathematics, USA, 455-460, 2010.
[13] C. Xu, On idempotency, involution and nilpotency of a linear combination of two
matrices, Linear Multilinear Algebra 63 (8), 1664-1677, 2015.
[14] C. Xu and R. Xu, Tripotency of a linear combination of two involutory matrices and
a tripotent matrix that mutually commute, Linear Algebra Appl. 437 (9), 2091-2109,
2012.
On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix
We characterize the involutiveness of the linear combinations of the form $a{\mathbf{A}} + b{\mathbf{B}}$ when $a,b$ are nonzero complex numbers, ${\mathbf{A}}$ is a quadratic $n \times n$ nonzero matrix and ${\mathbf{B}}$ is an arbitrary $n \times n$ nonzero matrix, under certain properties imposed on $\mathbf{A}$ and $\mathbf{B}$.
[1] C. Bu and Y. Zhou, Involutory and s+1 potency of linear combinations of a tripotent
matrix and an arbitrary matrix, J. Appl. Math. Inform. 29 (1–2), 485-495, 2011.
[2] X. Liu, J. Benítez and M. Zhang, Involutiveness of linear combinations of a quadratic
or tripotent matrix and an arbitrary matrix, Bull. Iranian Math. Soc. 42 (3), 595-610,
2016.
[3] H. Özdemir and T. Petik, On spectra of some matrices derived from two quadratic
matrices, Bull. Iranian Math. Soc. 39 (2), 225-238, 2013.
[4] H. Özdemir and M. Sarduvan, Notes on linear combinations of two tripotent, idempotent,
and involutive matrices that commute, An. Ştiint. Univ. “Ovidius" Constanta
Ser. Mat. 16, 83-90, 2008.
[5] T. Petik, H. Özdemir and J. Benítez, On the spectra of some combinations of two
generalized quadratic matrices, Appl. Math. Comput. 268, 978-990, 2015.
[6] T. Petik, M. Uç and H. Özdemir, Generalized quadraticity of linear combination of
two generalized quadratic matrices, Linear Multilinear Algebra 63 (12), 2430-2439,
2015.
[7] M. Sarduvan and H. Özdemir, On linear combinations of two tripotent, idempotent,
and involutive matrices, Appl. Math. Comput. 200 (1), 401-406, 2008.
[8] M. Sarduvan and N. Kalaycı, On idempotency of linear combinations of a quadratic
or a cubic matrix and an arbitrary matrix, Filomat 33 (10), 3161-3185, 2019.
[9] M. Tošić, On some linear combinations of commuting involutive and idempotent matrices,
Appl. Math. Comput. 233, 103-108, 2014.
[10] M. Uç, H. Özdemir and A.Y. Özban, On the quadraticity of linear combinations of
quadratic matrices, Linear Multilinear Algebra 63 (6), 1125-1137, 2015.
[11] M. Uç, T. Petik and H. Özdemir, The generalized quadraticity of linear combinations
of two commuting quadratic matrices, Linear Multilinear Algebra 64 (9), 1696-1715,
2016.
[12] Y. Wu, K–potent matrices–construction and applications in digital image encryption,
Recent Advances in Applied Mathematics, AMERICAN-MATH’10 Proceedings of
the 2010 American Conference on Applied Mathematics, USA, 455-460, 2010.
[13] C. Xu, On idempotency, involution and nilpotency of a linear combination of two
matrices, Linear Multilinear Algebra 63 (8), 1664-1677, 2015.
[14] C. Xu and R. Xu, Tripotency of a linear combination of two involutory matrices and
a tripotent matrix that mutually commute, Linear Algebra Appl. 437 (9), 2091-2109,
2012.
Kalaycı, N., & Sarduvan, M. (2021). On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix. Hacettepe Journal of Mathematics and Statistics, 50(4), 1012-1027. https://doi.org/10.15672/hujms.705784
AMA
Kalaycı N, Sarduvan M. On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix. Hacettepe Journal of Mathematics and Statistics. August 2021;50(4):1012-1027. doi:10.15672/hujms.705784
Chicago
Kalaycı, Nurgül, and Murat Sarduvan. “On Involutiveness of Linear Combinations of a Quadratic Matrix and an Arbitrary Matrix”. Hacettepe Journal of Mathematics and Statistics 50, no. 4 (August 2021): 1012-27. https://doi.org/10.15672/hujms.705784.
EndNote
Kalaycı N, Sarduvan M (August 1, 2021) On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix. Hacettepe Journal of Mathematics and Statistics 50 4 1012–1027.
IEEE
N. Kalaycı and M. Sarduvan, “On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 4, pp. 1012–1027, 2021, doi: 10.15672/hujms.705784.
ISNAD
Kalaycı, Nurgül - Sarduvan, Murat. “On Involutiveness of Linear Combinations of a Quadratic Matrix and an Arbitrary Matrix”. Hacettepe Journal of Mathematics and Statistics 50/4 (August 2021), 1012-1027. https://doi.org/10.15672/hujms.705784.
JAMA
Kalaycı N, Sarduvan M. On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix. Hacettepe Journal of Mathematics and Statistics. 2021;50:1012–1027.
MLA
Kalaycı, Nurgül and Murat Sarduvan. “On Involutiveness of Linear Combinations of a Quadratic Matrix and an Arbitrary Matrix”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 4, 2021, pp. 1012-27, doi:10.15672/hujms.705784.
Vancouver
Kalaycı N, Sarduvan M. On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix. Hacettepe Journal of Mathematics and Statistics. 2021;50(4):1012-27.