关于矩阵范数的4个等式

Four equations on matrix norm

给出了关于范数的4个等式,其中infμm axx≠0‖Ax‖μ/‖X‖μ=ρ(A)等价于infμ‖A‖=ρ(A),在一般文献中已给出,另外3个等式鲜见于其他文献,为了理论上的完善,本文给出这3个等式的证明。同时,本文还给出limk→+∞‖Ak‖1/k=ρ(A)的一个新的证明。

Four equations on matrix norm are given in this paper, infmax μ max x≠0 ｜｜Ax｜｜μ/｜｜X｜｜μ=ρ（A）.One of the equation is an equivalence of the equation inf inf μ ｜｜A｜｜=ρ（A） ,which is given in other articles. The other 3 equations are rarely seen in other articles. The proof of the 3 equations is provided for theoretical perfection. In addition, a new way of proof of ｜｜A^k｜｜^1/k（k→＋∞）is given.