“power(of a matrix)”的概念、定义、翻译、参考文献-科学参考

power(of a matrix)

Mathematics

If A is a square matrix, then the products AA, AAA, AAAA,… are defined, and these powers of A are denoted by A², A³, A⁴,…. For all positive integers p and q,

$(i) A^{p} A^{q} = A^{p + q} = A^{q} A^{p}, and (ii) {(A^{p})}^{q} = A^{p q} .$

By definition, A⁰ = I. Moreover, if A is invertible, then A², A³,…are invertible, and it can be shown that (A⁻¹)^p is the inverse of A^p, so that (A⁻¹)^p = (A^p)⁻¹. So either of these is denoted by A^−p. Thus, when A is invertible, A^p has been defined for all integers and properties (i) and (ii) above hold for all integers p and q.

单词	power(of a matrix)
释义	power(of a matrix) Mathematics If A is a square matrix, then the products AA, AAA, AAAA,… are defined, and these powers of A are denoted by A², A³, A⁴,…. For all positive integers p and q, $(i) A^{p} A^{q} = A^{p + q} = A^{q} A^{p}, and (ii) {(A^{p})}^{q} = A^{p q} .$ By definition, A⁰ = I. Moreover, if A is invertible, then A², A³,…are invertible, and it can be shown that (A⁻¹)^p is the inverse of A^p, so that (A⁻¹)^p = (A^p)⁻¹. So either of these is denoted by A^−p. Thus, when A is invertible, A^p has been defined for all integers and properties (i) and (ii) above hold for all integers p and q.
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