• Journal of applied mathematics & informatics
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• 제27권5_6호
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• pp.1493-1499
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• 2009

Let $A_1$ and $A_2$ be nonzero complex idempotent and tripotent matrix, respectively. Denote a linear combination of the two matrices by A = $c_1A_1$ + $c_2A_2$, where $c_1,\;c_2$ are nonzero complex scalars. In this paper, under an assumption of $A_1A_2$ = $A_2A_1$, we characterize all situations in which the linear combination is tripotent. A statistical interpretation of this tripotent problem is also pointed out. Moreover, In [2], Baksalary characterized all situations in which the above linear combination is idem-potent by using the property of decomposition of a tripotent matrix, i.e. if $A_2$ is tripotent, then $A_2$ = $B_1-B_2$, where $B^2_i=B_i$, i = 1, 2 and $B_1B_2=B_2B_1=0$. While in this paper, by utilizing a method different from the one used by Baksalary in [2], we prove the theorem 1 in [2] again.

• 대한수학회보
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• 제55권3호
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• pp.913-920
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• 2018

Let R be a 2-primal strongly 2-nil-clean ring. We prove that every square matrix over R is the sum of a tripotent and a nilpotent matrices. The similar result for rings of bounded index is proved. We thereby provide a large class of rings over which every matrix is the sum of a tripotent and a nilpotent matrices.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.485-495
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• 2011

Let $A_1$ and $A_2$ be $n{\times}n$ nonzero complex matrices, denote a linear combination of the two matrices by $A=c_1A_1+c_2A_2$, where $c_1$, $c_2$ are nonzero complex numbers. In this paper, we research the problem of the linear combinations in the general case. We give a sufficient and necessary condition for A is an involutive matrix and s+1-potent matrix, respectively, where $A_1$ is a tripotent matrix, with $A_1A_2=A_2A_1$. Then, using the results, we also give the sufficient and necessary conditions for the involutory of the linear combination A, where $A_1$ is a tripotent matrix, anti-idempotent matrix, and involutive matrix, respectively, and $A_2$ is a tripotent matrix, idempotent matrix, and involutive matrix, respectively, with $A_1A_2=A_2A_1$.

• 대한수학회논문집
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• 제10권1호
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• pp.15-21
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• 1995

• 호남수학학술지
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• 제46권3호
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• pp.349-364
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• 2024

We give necessary and sufficient conditions for tripotency of the linear combination of the form aQ + bA, under some certain conditions imposed on Q and A, where Q is a nonzero quadratic matrix, A is a nonzero arbitrary matrix and a, b are nonzero complex numbers. Moreover, some examples illustrating the main results are given.

• 대한수학회논문집
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• 제12권1호
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• pp.27-36
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• 1997

A matrix whose entries consist of the symbols +, -, 0 is called a sign-pattern matrix. We characterize the $n \times n$ irreducible sign-pattern matrices that are sign tripotent.